Radicals & Absolute Values - PSAT Math

1. What is the solution set of $|x-2|<3$ written as a compound inequality?

   $-1<x<5$. $-3 < x-2 < 3$ adds 2 to all parts: $-1 < x < 5$.

2. What is the quotient rule for radicals when $a\ge 0$ and $b>0$?

   $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. Quotient under radical can be split into quotient of radicals.

3. What is the value of $\sqrt{(-5)^2}$?

   $5$. $\sqrt{(-5)^2} = \sqrt{25} = 5 = |-5|$.

4. What is the value of $\frac{\sqrt{18}}{\sqrt{2}}$ in simplest form?

   $3$. $\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$.

5. What is the value of $\sqrt{\frac{1}{49}}$?

   $\frac{1}{7}$. Since $(\frac{1}{7})^2 = \frac{1}{49}$, this is the square root.

6. What is the simplified value of $\sqrt{12}\cdot\sqrt{3}$?

   $6$. Using the product rule: $\sqrt{12} \cdot \sqrt{3} = \sqrt{36} = 6$.

7. What is the value of $|{-17}|$?

   $17$. Absolute value makes negative numbers positive.

8. What is the value of $| -17 |$?

   $17$. Absolute value makes negative numbers positive.

9. What is the value of $|{-12}|$?

   $12$. Absolute value removes the negative sign.

10. What is the value of $\sqrt{(-4)^2}$?

    $4$. $(-4)^2 = 16$, so $\sqrt{16} = 4$.

11. What is the solution set of $|x|<5$?

    $-5<x<5$. $|x| < 5$ means $x$ is less than 5 units from zero.

12. What is the key product identity for conjugates $(a+\sqrt{b})(a-\sqrt{b})$?

    $(a+\sqrt{b})(a-\sqrt{b})=a^2-b$. Multiplying conjugates eliminates the radical terms.

13. What is the solution set of $|2x+1|=9$?

    $x=4$ or $x=-5$. $2x+1 = 9$ gives $x=4$; $2x+1 = -9$ gives $x=-5$.

14. What is the solution set of $\sqrt{x-1}=3$ over the reals?

    $x=10$. Square both sides: $x-1 = 9$, so $x = 10$.

15. What is the solution set of $|x-4|=3$?

    $x=7$ or $x=1$. $x - 4 = 3$ or $x - 4 = -3$, giving $x = 7$ or $x = 1$.

16. What is the product $\sqrt{3}\cdot\sqrt{12}$ simplified?

    $6$. $\sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6$.

17. What is the product rule for radicals: $\sqrt{a}\,\sqrt{b}$ for $a\ge 0$ and $b\ge 0$?

    $\sqrt{a}\,\sqrt{b}=\sqrt{ab}$. Radicals can be multiplied under one radical sign.

18. Solve the inequality for $x$: $|x-2|\ge 3$.

    $x\le -1$ or $x\ge 5$. $|x-2|\ge 3$ means $x-2\le -3$ or $x-2\ge 3$.

19. What is $\frac{1}{\sqrt{5}}$ with a rationalized denominator?

    $\frac{\sqrt{5}}{5}$. Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to get $\frac{\sqrt{5}}{5}$.

20. What is the simplified form of $\sqrt{12x^2}$ assuming $x \ge 0$?

    $2x\sqrt{3}$. Factor out perfect squares: $\sqrt{12x^2} = \sqrt{4 \cdot 3 \cdot x^2} = 2x\sqrt{3}$.

21. What is the solution set of $|2x+1|=9$?

    $x=4$ or $x=-5$. $2x + 1 = 9$ or $2x + 1 = -9$, solving gives $x = 4$ or $x = -5$.

22. What is the value of $|x-4|$ when $x=1$?

    $3$. $1 - 4 = -3$, and $|-3| = 3$.

23. What is $|{-17}|$?

    $17$. Absolute value removes the negative sign.

24. What is the simplified form of $\sqrt{ab}$ when $a\ge^0$ and $b\ge^0$?

    $\sqrt{ab}=\sqrt{a}\,\sqrt{b}$. Product rule: square root of a product equals product of square roots.

25. What is the solution set of $|x-3|=5$?

    $x=8$ or $x=-2$. $x-3 = 5$ gives $x=8$; $x-3 = -5$ gives $x=-2$.

26. What is the value of $\sqrt{a^2}$ for a real number $a$?

    $|a|$. The square root of a square gives the absolute value.

27. What is the solution set of $|x-2|\ge 4$ written with $\le$ or $\ge$?

    $x\le -2$ or $x\ge 6$. $|x - 2| \ge 4$ means $x - 2 \le -4$ or $x - 2 \ge 4$.

28. What is the value of $\sqrt{0.01}$?

    $0.1$. Since $0.1^2 = 0.01$, the square root of 0.01 is 0.1.

29. What is the value of $\sqrt{18}\cdot\sqrt{2}$?

    $6$. Using the product rule: $\sqrt{18} \cdot \sqrt{2} = \sqrt{36} = 6$.

30. What is the definition of $|x|$ in terms of $x$ and $-x$?

    $|x|$ is the distance from $0$; equals $x$ if $x \ge 0$, else $-x$. Absolute value removes the sign, making negatives positive.

31. What is the simplified form of $\sqrt{ab}$ for $a\ge^0$ and $b\ge^0$?

    $\sqrt{ab}=\sqrt{a}\,\sqrt{b}$. Product rule: square root of a product equals product of square roots.

32. What is the value of $\sqrt{\frac{1}{16}}$?

    $\frac{1}{4}$. Since $(\frac{1}{4})^2 = \frac{1}{16}$, this is the square root.

33. What is the solution set of $|x|\le 3$ written as an interval?

    $[-3,3]$. Distance from 0 at most 3 includes the endpoints.

34. What is the definition of absolute value $|x|$ written as a piecewise function?

    $|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}$. Absolute value returns the distance from zero, always non-negative.

35. What is the definition of $|x|$ in terms of cases?

    $|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value keeps positive values unchanged, makes negative values positive.

36. What is the value of $\sqrt{50}$ in simplest radical form?

    $5\sqrt{2}$. $50 = 25 \cdot 2 = 5^2 \cdot 2$, so $\sqrt{50} = 5\sqrt{2}$.

37. What is the value of $|{-12}|$?

    $12$. Absolute value makes negative numbers positive.

38. What is the simplified form of $\frac{5}{\sqrt{3}}$ with no radical in the denominator?

    $\frac{5\sqrt{3}}{3}$. Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$ to rationalize: $\frac{5\sqrt{3}}{3}$.

39. What is the domain of $\sqrt{x}$ in the real numbers?

    $x\ge^0$. Square roots of negative numbers are not real, so input must be non-negative.

40. What is the solution set of $|x|=7$?

    $x=7$ or $x=-7$. The values that are 7 units from zero are $\pm 7$.

41. What is the solution set of $|x-2|\le^5$?

    $-3\le x\le^7$. Values within 5 units of 2: $-5 \le x-2 \le 5$.

42. What is the definition of $\sqrt{a}$ for $a \ge 0$?

    $\sqrt{a}$ is the nonnegative number whose square is $a$. The square root gives the principal (nonnegative) root only.

43. What is the value of $|\!-7|$?

    $7$. Absolute value makes negative numbers positive: $|-7| = 7$.

44. What is the rationalized form of $\frac{3}{\sqrt{5}}$?

    $\frac{3\sqrt{5}}{5}$. Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to rationalize the denominator.

45. What is the value of $\sqrt{0.81}$?

    $0.9$. Since $(0.9)^2 = 0.81$.

46. What is the simplified form of $\sqrt{72}$?

    $6\sqrt{2}$. Factor out perfect squares: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

47. What is the simplified value of $\sqrt{a^2}$ for real $a$?

    $\sqrt{a^2}=|a|$. Square root of a squared value gives the absolute value.

48. What is $\sqrt{x^2}$ when $x<0$?

    $-x$. When $x < 0$, $|x| = -x$, and $\sqrt{x^2} = |x|$.

49. What is the value of $|{-x}|$ in simplest form?

    $|{-x}|=|x|$. Absolute value measures distance from zero, regardless of sign.

50. What is the simplified form of $\sqrt{\frac{9}{16}}$?

    $\frac{3}{4}$. Apply the quotient rule: $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$.

51. What is the value of $\sqrt{144}$?

    $12$. Since $12^2 = 144$.

52. What is the simplified form of $\sqrt{50x^2}$ for real $x$?

    $5|x|\sqrt{2}$. Factor: $50x^2 = 25 \cdot 2 \cdot x^2$, and $\sqrt{x^2} = |x|$.

53. What is the solution set of $|2x+1|=7$?

    $x=3$ or $x=-4$. Solve $2x+1 = 7$ and $2x+1 = -7$.

54. What is the distance between $a$ and $b$ on a number line in terms of absolute value?

    $|a-b|$. Distance is always positive, measured by absolute difference.

55. What is the simplified value of $|3-10|$?

    $7$. $3-10 = -7$, and $|-7| = 7$.

56. What is the value of $\sqrt{12}\cdot\sqrt{3}$?

    $6$. Apply product rule: $\sqrt{12 \cdot 3} = \sqrt{36} = 6$.

57. What is the value of $|x|^2$ in simplest form?

    $|x|^2=x^2$. Squaring eliminates the sign, so $|x|^2$ and $x^2$ are equivalent.

58. What is the definition of $\sqrt{a}$ for $a \ge 0$?

    $\sqrt{a}$ is the nonnegative number $r$ such that $r^2=a$. The square root gives the positive value whose square equals the radicand.

59. What is the product rule for radicals when $a\ge 0$ and $b\ge 0$?

    $\sqrt{ab}=\sqrt{a}\,\sqrt{b}$. Product under radical can be split into product of radicals.

60. What is the solution set of $|x|=7$?

    $x=7$ or $x=-7$. Absolute value equation has two solutions: positive and negative.

61. Identify the simplified form of $\sqrt{72}$.

    $6\sqrt{2}$. $\sqrt{72}=\sqrt{36\cdot 2}=\sqrt{36}\cdot\sqrt{2}=6\sqrt{2}$.

62. What is the product rule for radicals when $a \ge 0$ and $b \ge 0$?

    $\sqrt{ab}=\sqrt{a}\sqrt{b}$. The square root of a product equals the product of square roots.

63. What is the solution set of $|x|=5$?

    $x=\pm^5$. Both $x = 5$ and $x = -5$ satisfy $|x| = 5$.

64. What is the solution set of $|x|<5$ written as a compound inequality?

    $-5<x<5$. $|x| < 5$ means $x$ is between $-5$ and $5$.

65. What is the solution set of $|x|<3$?

    $-3<x<3$. All values within 3 units of zero.

66. What is the simplified form of $\sqrt{50}$?

    $5\sqrt{2}$. Factor out perfect squares: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.

67. What is the solution set of $|x|=9$?

    $x=9$ or $x=-9$. $|x| = 9$ means $x = 9$ or $x = -9$.

68. What is the solution set of $|2x+1|=7$?

    $x=3$ or $x=-4$. $2x + 1 = 7$ or $2x + 1 = -7$, giving $x = 3$ or $x = -4$.

69. What is the simplified form of $\sqrt{18x^2}$ for $x \ge 0$?

    $3x\sqrt{2}$. $\sqrt{18x^2} = \sqrt{9 \times 2 \times x^2} = 3|x|\sqrt{2} = 3x\sqrt{2}$ since $x \ge 0$.

70. What is the value of $\sqrt{x^2}$ for real $x$?

    $|x|$. $\sqrt{x^2} = |x|$ because the result must be nonnegative.

71. What is the solution set of $|x-4|=3$?

    $x=7$ or $x=1$. $x - 4 = 3$ or $x - 4 = -3$, giving $x = 7$ or $x = 1$.

72. What is the quotient $\frac{\sqrt{45}}{\sqrt{5}}$ simplified?

    $3$. $\frac{\sqrt{45}}{\sqrt{5}} = \sqrt{\frac{45}{5}} = \sqrt{9} = 3$.

73. What is the simplified value of $\frac{\sqrt{48}}{\sqrt{3}}$?

    $4$. Using the quotient rule: $\frac{\sqrt{48}}{\sqrt{3}} = \sqrt{\frac{48}{3}} = \sqrt{16} = 4$.

74. What is the definition of absolute value $|x|$?

    $|x|=x$ if $x\ge 0$ and $|x|=-x$ if $x<0$. Absolute value gives the distance from zero on the number line.

75. What is the value of $\sqrt{72}$ in simplest radical form?

    $6\sqrt{2}$. $72 = 36 \cdot 2 = 6^2 \cdot 2$, so $\sqrt{72} = 6\sqrt{2}$.

76. What is the simplified value of $\sqrt{(-5)^2}$?

    $5$. $(-5)^2 = 25$, and $\sqrt{25} = 5$ (principal root is positive).

77. What is the simplified form of $\frac{\sqrt{45}}{\sqrt{5}}$?

    $3$. Using the quotient rule: $\frac{\sqrt{45}}{\sqrt{5}} = \sqrt{\frac{45}{5}} = \sqrt{9} = 3$.

78. What is the value of $\sqrt{\frac{1}{16}}$?

    $\frac{1}{4}$. Since $(\frac{1}{4})^2 = \frac{1}{16}$, the square root is $\frac{1}{4}$.

79. What is the value of $\sqrt{0}$?

    $0$. Since $0^2 = 0$, the square root of zero is zero.

80. What is the solution set of $|x-2|=3$?

    $x=-1\text{ or }x=5$. $x - 2 = 3$ gives $x = 5$; $x - 2 = -3$ gives $x = -1$.

81. What is the value of $\sqrt{72}$ in simplest radical form?

    $6\sqrt{2}$. $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

82. What is the value of $\sqrt{16x^2}$ for real $x$?

    $4|x|$. $\sqrt{16x^2} = \sqrt{16} \cdot \sqrt{x^2} = 4|x|$.

83. What is the solution set of $|2x-5|=7$?

    $x=-1$ or $x=6$. Solve $2x-5 = 7$ and $2x-5 = -7$ to get both solutions.

84. What is the simplified value of $| -12 |$?

    $12$. Absolute value removes the negative sign.

85. What is the principal value of $\sqrt{81}$?

    $9$. Since $9^2 = 81$, the principal square root is $9$.

86. What is the value of $\sqrt{50}$ in simplest radical form?

    $5\sqrt{2}$. $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.

87. What is the solution set of $|x|\ge 3$ written using inequalities?

    $x\le -3$ or $x\ge 3$. Values at least 3 units from zero are outside $[-3, 3]$.

88. What is the solution set of $|x-4|=3$?

    $x=7$ or $x=1$. $x-4 = 3$ or $x-4 = -3$, giving $x = 7$ or $x = 1$.

89. What is the value of $|-7|$?

    $7$. Absolute value makes negative numbers positive.

90. What is the definition of $\sqrt{x}$ for real numbers $x$?

    $\sqrt{x}$ is the nonnegative real number $r$ with $r^2=x$. The principal square root is always nonnegative by definition.

91. What is the value of $\sqrt{18x^2}$ in simplest form for all real $x$?

    $3|x|\sqrt{2}$. $\sqrt{x^2} = |x|$, so $\sqrt{18x^2} = 3|x|\sqrt{2}$.

92. What is the solution set of $|x|=5$?

    $x=5$ or $x=-5$. The equation $|x| = 5$ means $x$ is 5 units from zero.

93. What is the solution set of $|x|<4$ written as an interval?

    $(-4,4)$. $|x| < 4$ means $-4 < x < 4$.

94. What is the simplified form of $\sqrt{\frac{48}{3}}$?

    $4$. $\sqrt{\frac{48}{3}} = \sqrt{16} = 4$.

95. What is the definition of $|x|$ in terms of a piecewise rule?

    $|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value keeps positive values unchanged, makes negatives positive.

96. What is the value of $\sqrt{49}$?

    $7$. Since $7^2 = 49$, the square root of 49 is 7.

97. What is the simplified value of $\sqrt{72}$?

    $6\sqrt{2}$. $72 = 36 \times 2 = 6^2 \times 2$, so $\sqrt{72} = 6\sqrt{2}$.

98. What is the solution set of $|x-3|=4$?

    $x=7$ or $x=-1$. Solve $x-3 = 4$ and $x-3 = -4$.

99. What is the simplified value of $\sqrt{\frac{49}{64}}$?

    $\frac{7}{8}$. $\sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} = \frac{7}{8}$.

100. What is the rule for a negative exponent with radicals: $a^{-\frac{m}{n}}$?

     $a^{-\frac{m}{n}}=\frac{1}{\sqrt[n]{a^m}}$. Negative exponent means reciprocal of the positive exponent form.

101. What is the solution set of $|x|<4$?

     $-4<x<4$. Numbers within 4 units of zero: $-4 < x < 4$.

102. What is the piecewise definition of $|x|$?

     $|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Keeps positive values unchanged, makes negative values positive.

103. What is the value of $\sqrt{49}$?

     $7$. Since $7^2 = 49$, the principal square root of 49 is 7.

104. What is the simplified form of $\sqrt{a^2}$ for real $a$?

     $|a|$. The square root of a squared term gives the absolute value.

105. What is the simplified value of $3\sqrt{8}$?

     $6\sqrt{2}$. $3\sqrt{8} = 3\sqrt{4 \times 2} = 3 \times 2\sqrt{2} = 6\sqrt{2}$.

106. What is the definition of $\sqrt{x}$ for $x\ge 0$?

     $\sqrt{x}$ is the nonnegative number whose square is $x$. The principal square root is always nonnegative by definition.

107. What is the value of $\sqrt{49}$?

     $7$. Since $7^2 = 49$, the square root of 49 is 7.

108. What is the simplified form of $\sqrt{18x^2}$ for real $x$?

     $3\sqrt{2}\,|x|$. $\sqrt{18x^2} = \sqrt{9 \cdot 2 \cdot x^2} = 3\sqrt{2}\,|x|$ since $\sqrt{x^2} = |x|$.

109. What is the solution set of $|x|=7$?

     $x=7$ or $x=-7$. Absolute value equation splits into two cases: $x = 7$ or $x = -7$.

110. What is the principal square root of $a^2$ for real $a$?

     $\sqrt{a^2}=|a|$. The principal square root always returns a non-negative value.

111. What is the definition of $|x|$ in terms of $x$ and $-x$?

     $|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value removes the sign, making negative values positive.

112. What is $\sqrt{(-3)^2}$?

     $3$. $(-3)^2 = 9$, and $\sqrt{9} = 3$ (always nonnegative).

113. What is the simplified form of $\sqrt{\frac{a}{b}}$ for $a \ge 0$ and $b>0$?

     $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. The square root of a quotient equals the quotient of square roots.

114. What is the value of $\frac{\sqrt{48}}{\sqrt{3}}$ in simplest form?

     $4$. Apply quotient rule: $\sqrt{\frac{48}{3}} = \sqrt{16} = 4$.

115. What is the definition of absolute value, $|x|$, written as a piecewise expression?

     $|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}$. Returns the value itself if non-negative, or its opposite if negative.

116. What is the simplified form of $\sqrt{ab}$ for $a \ge 0$ and $b \ge 0$?

     $\sqrt{ab}=\sqrt{a}\sqrt{b}$. The square root of a product equals the product of square roots.

117. What is the distance between $-3$ and $5$ written using absolute value?

     $|-3-5|=8$. Distance is $|5-(-3)| = |8| = 8$.

118. What is the principal square root of $a^2$ for real $a$?

     $\sqrt{a^2}=|a|$. The principal square root is always non-negative, so we need absolute value.

119. What is the simplified form of $\sqrt{x^2}$ for real $x$?

     $|x|$. The square root of a squared variable equals its absolute value.

120. What is $\sqrt{a^2}$ in terms of $|a|$ for real $a$?

     $\sqrt{a^2}=|a|$. Square root of a square always gives the absolute value.

121. What is the solution set of $\sqrt{x+4}<5$ over the reals?

     $-4\le x<21$. Need $x+4 \ge 0$ and $x+4 < 25$.

122. What is the definition of $\sqrt{a}$ for $a \ge 0$?

     $\sqrt{a}$ is the nonnegative number whose square is $a$. The square root gives the positive value that, when squared, equals the original number.

123. What is the solution set of $|x|<5$ written as a compound inequality?

     $-5<x<5$. Values within 5 units of zero: $-5 < x < 5$.

124. What is the value of $\sqrt{\frac{49}{64}}$?

     $\frac{7}{8}$. Apply $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ to get $\frac{7}{8}$.

125. What is the value of $\sqrt{\frac{49}{64}}$?

     $\frac{7}{8}$. Apply the quotient rule: $\sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} = \frac{7}{8}$.

126. What is the value of $\sqrt{12}+\sqrt{27}$ in simplest form?

     $5\sqrt{3}$. $\sqrt{12} = 2\sqrt{3}$ and $\sqrt{27} = 3\sqrt{3}$, sum is $5\sqrt{3}$.

127. What is the simplified form of $\sqrt{16x^4}$ for real $x$?

     $4x^2$. $\sqrt{16x^4} = \sqrt{16} \cdot \sqrt{(x^2)^2} = 4x^2$ since $x^2 \geq 0$.

128. What is the meaning of $\sqrt{a^2b}$ when $a$ is a real number and $b\ge 0$?

     $\sqrt{a^2b}=|a|\sqrt{b}$. Factor out perfect squares and use absolute value for variables.

129. What is the solution set of $|x|\ge^4$?

     $x\le-4$ or $x\ge^4$. Values at least 4 units from zero.

130. What is the quotient rule for radicals when $a \ge 0$ and $b>0$?

     $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. The square root of a quotient equals the quotient of square roots.

131. What is the simplified form of $\sqrt{\frac{45}{5}}$?

     $3$. Simplify inside first: $\frac{45}{5} = 9$, then $\sqrt{9} = 3$.

132. What is the value of $\sqrt{72}$ in simplest radical form?

     $6\sqrt{2}$. Factor out perfect square: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

133. What is the solution set of $|x|=9$?

     $x=-9$ or $x=9$. Absolute value equations have two solutions: positive and negative.

134. What is the value of $\sqrt{\frac{1}{16}}$?

     $\frac{1}{4}$. Since $(\frac{1}{4})^2 = \frac{1}{16}$, this is the square root.

135. What is the value of $\sqrt{x^2}$ if $x\le 0$?

     $-x$. When $x \le 0$, $|x| = -x$, so $\sqrt{x^2} = -x$.

136. What is the solution set of $|2x+1|=7$?

     $x=3\text{ or }x=-4$. $2x + 1 = 7$ gives $x = 3$; $2x + 1 = -7$ gives $x = -4$.

137. What is the value of $\sqrt{72}$ in simplest radical form?

     $6\sqrt{2}$. $72 = 36 \times 2 = 6^2 \times 2$, so factor out the perfect square.

138. What is the value of $\sqrt{72}$ in simplest radical form?

     $6\sqrt{2}$. $72 = 36 \cdot 2 = 6^2 \cdot 2$, so $\sqrt{72} = 6\sqrt{2}$.

139. What is the value of $\sqrt{50}$ in simplest radical form?

     $5\sqrt{2}$. $50 = 25 \cdot 2 = 5^2 \cdot 2$, so $\sqrt{50} = 5\sqrt{2}$.

140. What is the simplified form of $\sqrt{18x^2}$ for $x<0$?

     $-3x\sqrt{2}$. $\sqrt{18x^2} = 3|x|\sqrt{2} = 3(-x)\sqrt{2} = -3x\sqrt{2}$ since $x < 0$.

141. What is the value of $\sqrt{(-5)^2}$?

     $5$. $\sqrt{(-5)^2} = \sqrt{25} = 5$ (principal root is positive).

142. What is the value of $|{-7}|$?

     $7$. Absolute value of $-7$ is its distance from zero: $7$.

143. What is the conjugate of a binomial radical expression $a+\sqrt{b}$?

     $a-\sqrt{b}$. Conjugate changes the sign of the radical term.

144. What is the solution set of $|x|=5$?

     $x=5$ or $x=-5$. The values that are 5 units from zero on the number line.

145. What is the result of multiplying conjugates: $(a+\sqrt{b})(a-\sqrt{b})$?

     $a^2-b$. Difference of squares pattern: $(a)^2 - (\sqrt{b})^2 = a^2 - b$.

146. What is the solution set of $|x-3|<2$?

     $1<x<5$. $|x-3| < 2$ means $x$ is within 2 units of 3.

147. What is $\sqrt{x^2}$ equal to for all real $x$?

     $|x|$. Square root of a square always gives the absolute value.

148. Rationalize the denominator: $\frac{3}{\sqrt{5}}$.

     $\frac{3\sqrt{5}}{5}$. Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to eliminate radical from denominator.

149. What is the value of $\sqrt{50}$ in simplest radical form?

     $5\sqrt{2}$. Factor out perfect square: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.

150. What is the simplified form of $\sqrt{(3x)^2}$ for real $x$?

     $3|x|$. $\sqrt{(3x)^2} = \sqrt{9x^2} = 3|x|$.

151. What is the definition of $\sqrt{x}$ for $x\ge 0$?

     $\sqrt{x}$ is the nonnegative number $r$ such that $r^2=x$. The square root gives the principal (nonnegative) root only.

152. Solve for $x$: $|x|=9$.

     $x=9$ or $x=-9$. $|x|=9$ means $x$ is $9$ units from zero in either direction.

153. Identify the simplified form of $\sqrt{50}$.

     $5\sqrt{2}$. $\sqrt{50}=\sqrt{25\cdot 2}=\sqrt{25}\cdot\sqrt{2}=5\sqrt{2}$.

154. What is the solution set of $|x+1|\ge 4$ written as two inequalities?

     $x\le -5$ or $x\ge 3$. $x+1 \le -4$ or $x+1 \ge 4$ gives $x \le -5$ or $x \ge 3$.

155. What is the value of $\sqrt{x^2}$ for real $x$?

     $|x|$. Squaring and taking square root gives absolute value.

156. What is the solution set of $|x|=9$?

     $x=9$ or $x=-9$. Absolute value equation gives two solutions: $x = 9$ and $x = -9$.

157. What is the principal value of $\sqrt{49}$?

     $7$. Since $7^2 = 49$, and we take the positive root.

158. What is the value of $|5-12|$?

     $7$. $5 - 12 = -7$, and $|-7| = 7$.

159. What is the value of $\sqrt{0.0009}$?

     $0.03$. Since $0.03^2 = 0.0009$, the square root is 0.03.

160. What is the solution set of $|x-2|\le 3$ written as an interval?

     $[-1,5]$. $-3 \le x-2 \le 3$ gives $-1 \le x \le 5$.

161. What is the value of $\sqrt{12}\cdot\sqrt{3}$ in simplest form?

     $6$. $\sqrt{12} \cdot \sqrt{3} = \sqrt{36} = 6$ by the product rule.

162. What is the solution set of $|x+1|>3$ written as a union of intervals?

     $(-\infty,-4)\cup(2,\infty)$. $x + 1 < -3$ or $x + 1 > 3$, so $x < -4$ or $x > 2$.

163. What is the value of $\sqrt{18x^2}$ in simplest form for $x\ge 0$?

     $3x\sqrt{2}$. $\sqrt{18x^2} = \sqrt{9 \cdot 2 \cdot x^2} = 3x\sqrt{2}$ when $x \ge 0$.

164. What is the value of $\sqrt{49}$?

     $7$. Since $7^2 = 49$.

165. What is the solution set of $|x-2|\le 6$ written as an interval?

     $[-4,8]$. $-6 \le x - 2 \le 6$, so $-4 \le x \le 8$.

166. What is the simplified form of $\sqrt{k^2}$ for real $k$?

     $\sqrt{k^2}=|k|$. Principal square root is always non-negative, requiring absolute value for real $k$.

167. What is the value of $\frac{\sqrt{48}}{\sqrt{3}}$ in simplest form?

     $4$. $\frac{\sqrt{48}}{\sqrt{3}} = \sqrt{\frac{48}{3}} = \sqrt{16} = 4$.

168. What is the solution set of $|2x+1|=5$?

     $x=2$ or $x=-3$. $2x+1 = 5$ or $2x+1 = -5$, solving gives $x = 2$ or $x = -3$.

169. What is the product rule for radicals: $\sqrt{a}\sqrt{b}$ (for $a\ge 0$, $b\ge 0$)?

     $\sqrt{a}\sqrt{b}=\sqrt{ab}$. Radicals can be multiplied under one radical sign.

170. What is the value of $\sqrt{0.81}$?

     $0.9$. Since $0.9^2 = 0.81$, this is the square root.

171. What is the value of $\sqrt{\frac{9}{16}}$?

     $\frac{3}{4}$. $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$.

172. What is the solution set of $|x|\ge 6$?

     $x\le -6$ or $x\ge 6$. Numbers at least 6 units from zero in either direction.

173. What is the simplified form of $\sqrt{25y^2}$ for real $y$?

     $5|y|$. $\sqrt{25y^2} = \sqrt{25} \cdot \sqrt{y^2} = 5|y|$ since $\sqrt{y^2} = |y|$.

174. What is the solution set of $|x-4|=9$?

     $x=13$ or $x=-5$. $x-4 = 9$ gives $x = 13$; $x-4 = -9$ gives $x = -5$.

175. What is the rule for converting a rational exponent to a radical: $a^{\frac{m}{n}}$?

     $a^{\frac{m}{n}}=\sqrt[n]{a^m}$. Rational exponent $\frac{m}{n}$ means the $n$th root of $a^m$.

176. What is the solution set of $|x+1|\ge^2$ written in interval notation?

     $(-\infty,-3]\cup[1,\infty)$. $|x + 1| \ge 2$ means $x + 1 \le -2$ or $x + 1 \ge 2$, so $x \le -3$ or $x \ge 1$.

177. What is the solution set of $|2x+1|=5$?

     $x=2$ or $x=-3$. $2x+1 = 5$ gives $x = 2$; $2x+1 = -5$ gives $x = -3$.

178. What is the value of $|-12|$?

     $12$. Absolute value removes the negative sign: $|-12| = 12$.

179. What is the solution set of $|x-4|=3$?

     $x=7$ or $x=1$. $x-4 = 3$ or $x-4 = -3$, giving $x = 7$ or $x = 1$.

180. What is the solution set of $|2x-1|\le 3$?

     $-1\le x\le 2$. $-3 \le 2x-1 \le 3$ gives $-2 \le 2x \le 4$, so $-1 \le x \le 2$.

181. What is the simplified form of $\sqrt{72}$?

     $6\sqrt{2}$. Factor out perfect squares: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

182. What is the simplified form of $\sqrt{\frac{a}{b}}$ for $a\ge^0$ and $b>0$?

     $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. Quotient rule: square root of a quotient equals quotient of square roots.

183. What is the solution set of $|x-3|=5$?

     $x=8$ or $x=-2$. $x-3 = 5$ gives $x=8$; $x-3 = -5$ gives $x=-2$.

184. What is the solution set of $|x|<4$ written as an interval?

     $(-4,4)$. Distance from 0 less than 4 means between -4 and 4.

185. What is the definition of $|x|$ for a real number $x$?

     $|x|=x$ if $x\ge 0$, and $|x|=-x$ if $x<0$. Absolute value removes the sign, making negative values positive.

186. What is the domain restriction for $\sqrt{x}$ over the real numbers?

     $x\ge^0$. Square roots of negative numbers are not real.

187. What is the solution set of $|x+1|>2$ written as a union of intervals?

     $(-\infty,-3)\cup(1,\infty)$. $x+1 > 2$ or $x+1 < -2$ gives $x > 1$ or $x < -3$.

188. What is the simplified form of $\sqrt{\frac{50}{2}}$?

     $5$. $\sqrt{\frac{50}{2}} = \sqrt{25} = 5$.

189. What is the value of $|3-8|$?

     $5$. $3 - 8 = -5$, and $|-5| = 5$.

190. What is the value of $| -17 |$?

     $17$. Absolute value makes negative numbers positive.

191. What is the meaning of $\sqrt{x^2}$ for a real number $x$?

     $\sqrt{x^2}=|x|$. Square root undoes squaring but always gives nonnegative result.

192. What is the simplified form of $\sqrt{50}$?

     $5\sqrt{2}$. Factor out perfect squares: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.

193. What is the rationalized form of $\frac{2}{3-\sqrt{5}}$?

     $\frac{3+\sqrt{5}}{2}$. Multiply by conjugate $\frac{3+\sqrt{5}}{3+\sqrt{5}}$ to rationalize.

194. What is the solution set of $|x-2|\le 4$ written as a compound inequality?

     $-2\le x\le 6$. $-4 \le x - 2 \le 4$, adding 2 gives $-2 \le x \le 6$.

195. What is the distance between $(1,2)$ and $(4,6)$?

     $5$. Use distance formula: $\sqrt{(4-1)^2+(6-2)^2} = \sqrt{9+16} = 5$.

196. Solve the inequality for $x$: $|x|<5$.

     $-5<x<5$. $|x|<5$ means $x$ is within $5$ units of zero.

197. What is the value of $\sqrt{49}$?

     $7$. Since $7^2 = 49$, the square root of 49 is 7.

198. What is the definition of $\sqrt{a}$ for $a\ge 0$ in real numbers?

     $\sqrt{a}$ is the nonnegative number whose square is $a$. The principal square root is always nonnegative by definition.

199. Solve for $x$: $|x-4|=6$.

     $x=10$ or $x=-2$. $|x-4|=6$ means $x-4=6$ or $x-4=-6$.

200. What is the solution set of $|x-3|=5$?

     $x=8$ or $x=-2$. $x - 3 = 5$ or $x - 3 = -5$, giving two solutions.

201. What is the quotient rule for radicals: $\frac{\sqrt{a}}{\sqrt{b}}$ (for $a\ge 0$, $b>0$)?

     $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$. Radicals can be divided under one radical sign when $b > 0$.

202. What is the solution set of $|x|<4$ written as a compound inequality?

     $-4<x<4$. $|x| < 4$ means $x$ is between $-4$ and $4$.

203. What is the definition of $\sqrt{a}$ for $a\ge 0$ (principal square root)?

     $\sqrt{a}$ is the nonnegative number $r$ such that $r^2=a$. The principal square root is always nonnegative by definition.

204. What is the solution set of $|2x|=10$?

     $x=5$ or $x=-5$. $2x = 10$ or $2x = -10$, so $x = 5$ or $x = -5$.

205. What is the definition of absolute value $|x|$?

     $|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value gives distance from zero on the number line.

206. What is the rationalized form of $\frac{5}{\sqrt{3}}$?

     $\frac{5\sqrt{3}}{3}$. Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$ to get $\frac{5\sqrt{3}}{3}$.

207. What is the value of $\sqrt{0.81}$?

     $0.9$. Since $0.9^2 = 0.81$, the square root is 0.9.

208. What is the value of $\sqrt{50}$ in simplest radical form?

     $5\sqrt{2}$. $50 = 25 \times 2 = 5^2 \times 2$, so factor out the perfect square.

209. What is the simplified form of $\sqrt{16x^2}$ for real $x$?

     $4|x|$. $\sqrt{16x^2} = \sqrt{16} \cdot \sqrt{x^2} = 4|x|$.

210. What is the simplified value of $\sqrt{50}$?

     $5\sqrt{2}$. $50 = 25 \times 2 = 5^2 \times 2$, so $\sqrt{50} = 5\sqrt{2}$.

211. What is the simplified form of $\sqrt{\frac{a}{b}}$ when $a\ge^0$ and $b>0$?

     $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. Quotient rule: square root of a fraction equals fraction of square roots.

212. What is the value of $|x|$ when $x=-4$?

     $4$. Absolute value of -4 is 4 (removes negative sign).

213. What is $(\sqrt{5})^2$?

     $5$. Squaring a square root returns the original value.

214. What is the value of $|-12|$?

     $12$. Absolute value makes negative numbers positive.

215. What is the value of $|-12|$?

     $12$. Absolute value removes the negative sign.

216. What is the value of $\sqrt{x^2}$ in terms of absolute value?

     $\sqrt{x^2}=|x|$. Square root of a square gives the absolute value, ensuring non-negative result.

217. What is the value of $\sqrt{0.0009}$?

     $0.03$. Since $(0.03)^2 = 0.0009$.

218. What is the definition of $|x|$ in terms of distance on the number line?

     $|x|$ is the distance from $x$ to $0$. Absolute value measures how far a number is from zero.

219. What is the definition of absolute value $|x|$?

     $|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value returns the distance from zero, always nonnegative.

220. What is the definition of $\sqrt{a}$ for $a\ge 0$?

     $\sqrt{a}$ is the nonnegative number $r$ such that $r^2=a$. The square root gives the positive value whose square equals $a$.

221. What is the value of $|3-10|$?

     $7$. $3-10=-7$, and $|-7|=7$.

222. What is the solution set of $|x-2|\le 5$?

     $-3\le x\le 7$. $-5 \le x-2 \le 5$, so $-3 \le x \le 7$.

223. What is the principal square root of $a$ for $a\ge 0$ (including its sign)?

     $\sqrt{a}\ge 0$. Principal square root is always non-negative by definition.

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