Radical & Exponential Functions - ACT Math

1. Find $x$ if $x^3 = 8$.

   2. Taking the cube root of both sides gives $x = 2$.

2. What is $3^4$?

   81. $3$ raised to the $4$th power equals $81$.

3. What is the base in an exponential function $y = ab^x$?

   $b$. The base determines the growth or decay factor.

4. Express $\text{√}(x^2)$ in simplest form.

   $|x|$. Square root of $x^2$ is the absolute value of $x$.

5. Simplify $\text{√}16$.

   4. The positive number that when squared equals 16.

6. Evaluate $(-2)^3$.

   -8. Cube the negative number: $(-2) \times (-2) \times (-2)$.

7. Identify the domain of $f(x)=\sqrt{x-5}$.

   $x\ge 5$. Expression under square root must be non-negative: $x-5 \ge 0$.

8. What is the definition of a zero exponent for $a \ne 0$, $a^0$?

   $1$. Any nonzero number raised to the power of zero equals 1.

9. What is the domain restriction for $f(x)=\sqrt{g(x)}$ over the reals?

   $g(x)\ge 0$. Expression under square root must be non-negative.

10. Identify the simplified value of $\frac{5^7}{5^2}$.

    $5^5$. Use quotient rule: $5^{7-2} = 5^5$.

11. What is the simplification of $\sqrt{\frac{a}{b}}$ for $a \ge 0$ and $b>0$?

    $\frac{\sqrt{a}}{\sqrt{b}}$. Square root of quotient equals quotient of square roots.

12. What is the horizontal asymptote of $f(x)=a^x$ for $a>0$ and $a\ne 1$?

    $y=0$. Exponential functions approach zero but never reach it.

13. What is the meaning of $a^{\frac{m}{n}}$ for $a \ge 0$ and integers $m,n>0$?

    $\sqrt[n]{a^m}$. Take the $n$th root of $a$ raised to the $m$th power.

14. State the formula for exponential decay.

    $y = a(1 - r)^t$. Where $a$ is initial value, $r$ is decay rate, $t$ is time.

15. Simplify: $\frac{\text{square root of } 16}{\text{square root of } 4}$.

    2. $\frac{\sqrt{16}}{\sqrt{4}} = \frac{4}{2} = 2$.

16. Identify the simplified value of $\sqrt{72}$ in simplest radical form.

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

17. What is the solution form for $a^x=a^k$ when $a>0$ and $a\ne 1$?

    $x=k$. Equal bases means equal exponents for exponential equations.

18. What is the value of $\text{√}1$?

    1. The square root of 1 is always 1.

19. State the value of $4^0$.

    1. Any non-zero number raised to the power $0$ equals $1$.

20. What is the value of $2^{-3}$?

    $\frac{1}{8}$. Negative exponent means reciprocal: $\frac{1}{2^3} = \frac{1}{8}$.

21. What is the definition of a rational exponent, $a^{\frac{1}{n}}$ for $a \ge 0$?

    $\sqrt[n]{a}$. Fractional exponent $\frac{1}{n}$ equals the $n$th root.

22. What is the square root of 81?

    9. $\sqrt{81} = 9$ since $9 \times 9 = 81$.

23. Identify the domain of $f(x)=\sqrt{\frac{x+1}{x-2}}$.

    $(-\infty,-1]\cup(2,\infty)$. Need $\frac{x+1}{x-2} \ge 0$ with $x \ne 2$.

24. Identify the simplified value of $2^3\cdot 2^5$.

    $2^8$. Use product rule: $2^{3+5} = 2^8$.

25. What is the value of $3^{-2}$?

    $\frac{1}{9}$. Negative exponents mean reciprocal: $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

26. What is the exponent rule for a quotient raised to a power, $\left(\frac{a}{b}\right)^n$?

    $\frac{a^n}{b^n}$. Distribute the exponent to numerator and denominator separately.

27. Simplify: $\frac{\text{√}50}{\text{√}2}$

    $5$. Use $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ to get $\sqrt{25} = 5$.

28. Simplify $\text{square root of } 100$.

    10. $\sqrt{100} = 10$ since $10^2 = 100$.

29. What is the value of $8^0$?

    1. Any non-zero number raised to the power $0$ equals $1$.

30. Identify the simplified value of $\sqrt{18x^2}$ for real $x$.

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

31. Find the value of $5^0$.

    1. Any non-zero number raised to the power $0$ equals $1$.

32. What is $2^3$?

    8. Multiply 2 by itself 3 times: $2 \times 2 \times 2$.

33. What is the definition of a negative exponent, $a^{-n}$ for $a \ne 0$?

    $\frac{1}{a^n}$. Negative exponent means reciprocal with positive exponent.

34. Rationalize the denominator: $\frac{3}{2\sqrt{5}}$.

    $\frac{3\sqrt{5}}{10}$. Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to eliminate radical in denominator.

35. What is the value of $10^0$?

    1. Any non-zero number raised to the power $0$ equals $1$.

36. Simplify $(\text{√}3)^2$.

    3. Squaring a square root returns the original number.

37. Simplify: $\text{square root of } 25$.

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

38. State the formula for exponential growth.

    $y = a(1 + r)^t$. Where $a$ is initial value, $r$ is growth rate, $t$ is time.

39. Simplify: $\text{square root of } 144$.

    12. $\sqrt{144} = 12$ since $12^2 = 144$.

40. What is $0.5$ as a power of $2$?

    $2^{-1}$. Negative exponent means reciprocal: $\frac{1}{2} = 2^{-1}$.

41. Find $x$ if $x^2 = 36$.

    6 or -6. Taking the square root of both sides gives $x = \pm 6$.

42. Simplify the expression $x^{\frac{2}{3}} \times x^{\frac{1}{3}}$.

    $x$. Add exponents when multiplying: $\frac{2}{3} + \frac{1}{3} = 1$, so $x^1 = x$.

43. What is $1$ as a power of any nonzero number?

    $a^0$. Zero exponent property: any nonzero base to power 0 equals 1.

44. What is the product rule for exponents with the same base, $a^m \cdot a^n$?

    $a^{m+n}$. Add exponents when multiplying powers with the same base.

45. Simplify: $\text{cube root of } 64$.

    4. $\sqrt[3]{64} = 4$ since $4^3 = 64$.

46. Convert $\sqrt{49}$ to an exponential form.

    $49^{\frac{1}{2}}$. Square root as fractional exponent with denominator 2.

47. Simplify: $\text{cube root of } 125$.

    5. $\sqrt[3]{125} = 5$ since $5^3 = 125$.

48. Simplify: $\text{square root of } 81$.

    9. $\sqrt{81} = 9$ since $9^2 = 81$.

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

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

50. Simplify: $\frac{\text{square root of } 64}{\text{square root of } 16}$.

    2. $\frac{\sqrt{64}}{\sqrt{16}} = \frac{8}{4} = 2$.

51. State the value of $2^5$.

    32. $2$ multiplied by itself $5$ times equals $32$.

52. State the exponential form of 100.

    $10^2$. $100 = 10 \times 10 = 10^2$.

53. What is the definition of a zero exponent for $a \neq 0$, $a^0$?

    1. Any nonzero number raised to the power of zero equals $1$.

54. What is the definition of a negative exponent, $a^{-n}$ for $a e 0$?

    $\frac{1}{a^n}$. Negative exponent means reciprocal with positive exponent.

55. Simplify $(\sqrt{3})^2$.

    $3$. Squaring a square root returns the original number.

56. What is the exponent in $7^2$?

    2. The exponent is the power to which the base is raised.

57. What is the domain restriction for $f(x)=a^{g(x)}$ when $a>0$ and $a\ne 1$?

    Domain is all real $x$. Positive base allows any real number as exponent.

58. What is the exponent rule for a product raised to a power, $(ab)^n$?

    $a^n b^n$. Distribute the exponent to each factor in the product.

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

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

60. What is $4^3$?

    64. $4$ cubed means $4 \times 4 \times 4 = 64$.

61. What is the inverse relationship between $y=a^x$ and logarithms for $a>0$, $a\ne 1$?

    $x=\log_a(y)$. Logarithm is the inverse operation of exponentiation.

62. State the property of exponents for dividing like bases.

    $a^m / a^n = a^{m-n}$. When dividing powers with same base, subtract the exponents.

63. What is the $y$-intercept of $f(x)=a^x$ for $a>0$?

    $(0,1)$. When $x=0$, any positive base equals 1.

64. What is the reciprocal of $3^{-2}$?

    $3^2$ or $9$. Negative exponent reciprocal: $\frac{1}{3^{-2}} = 3^2 = 9$.

65. What is the quotient rule for exponents with the same base, $\frac{a^m}{a^n}$?

    $a^{m-n}$. Subtract exponents when dividing powers with the same base.

66. Which condition makes $f(x)=a^x$ a decreasing function?

    $0<a<1$. Base between 0 and 1 creates exponential decay.

67. Simplify: $\frac{\sqrt{50}}{\sqrt{2}}$

    $5$. Use $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ to get $\sqrt{25} = 5$.

68. Identify the simplified value of $\sqrt{50}$ in simplest radical form.

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

69. What is the power rule for a power of a power, $(a^m)^n$?

    $a^{mn}$. Multiply exponents when raising a power to a power.

70. Identify the simplified value of $(3^2)^4$.

    $3^8$. Use power rule: $(3^2)^4 = 3^{2 \cdot 4} = 3^8$.

71. What is the square of 8?

    64. $8^2 = 8 \times 8 = 64$.

72. What is the fourth root of 81?

    3. The number that when raised to the 4th power equals 81.

73. What is $\text{√}(100)$?

    10. The positive number that when squared equals 100.

74. Find $x$ if $x^2 = 49$.

    7 or -7. Taking the square root of both sides gives $x = \pm 7$.

75. What is the exponential form of $1000$?

    $10^3$. $1000 = 10 \times 10 \times 10 = 10^3$.

76. Simplify $\frac{\text{√}75}{\text{√}3}$.

    $5$. Use $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ to get $\sqrt{25} = 5$.

77. Simplify: $\text{square root of } 36$.

    6. $\sqrt{36} = 6$ since $6^2 = 36$.

78. What is the square root of 49?

    7. The positive number that when multiplied by itself equals 49.

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

    $a^2-b$. Multiplying conjugates eliminates the radical term.

80. What is the value of $7^1$?

    7. Any number raised to the power $1$ equals itself.

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

    $a-\sqrt{b}$. Change the sign of the radical term to form conjugate.

82. Simplify $x^{\frac{1}{2}} \times x^{\frac{1}{2}}$.

    $x$. Add exponents when multiplying: $\frac{1}{2} + \frac{1}{2} = 1$, so $x^1 = x$.

83. Evaluate the expression $81^{\frac{1}{2}}$.

    9. Fractional exponent $\frac{1}{2}$ means square root: $\sqrt{81} = 9$.

84. What is the simplified value of $4^{\frac{3}{2}}$?

    $8$. $4^{3/2} = (2^2)^{3/2} = 2^3 = 8$.

85. Calculate: $2^4$.

    16. $2$ raised to the $4$th power equals $16$.

86. What is the simplified value of $16^{-\frac{1}{2}}$?

    $\frac{1}{4}$. $16^{-1/2} = \frac{1}{16^{1/2}} = \frac{1}{4}$.

87. Evaluate $4^{-2}$.

    $\frac{1}{16}$. Negative exponent means reciprocal: $\frac{1}{4^2} = \frac{1}{16}$.

88. What is the cube of 2?

    8. $2$ cubed means $2 \times 2 \times 2 = 8$.

89. What is the simplification of $\sqrt{ab}$ for $a \ge 0$ and $b \ge 0$?

    $\sqrt{a}\sqrt{b}$. Square root of product equals product of square roots.

90. What is the range of $f(x)=a^x$ for $a>0$ and $a\ne 1$?

    $(0,\infty)$. Exponential functions always output positive values.

91. Identify the domain of $f(x)=\sqrt{9-x^2}$.

    $-3\le x\le 3$. Need $9-x^2 \ge 0$, so $x^2 \le 9$.

92. What is the fourth root of 16?

    2. $\sqrt[4]{16} = 2$ since $2^4 = 16$.

93. Rationalize the denominator: $\frac{4}{1+\sqrt{2}}$.

    $4(\sqrt{2}-1)$. Multiply by conjugate $\frac{\sqrt{2}-1}{\sqrt{2}-1}$ to rationalize.

94. State the formula for the exponential function.

    $f(x) = a \times b^x$. Standard form where $a$ is initial value and $b$ is growth factor.

95. Identify the base in $5^3$.

    5. The base is the number being raised to a power.

96. What is the square of 9?

    81. $9^2 = 9 \times 9 = 81$.

97. State the property of exponents for multiplying like bases.

    $a^m \times a^n = a^{m+n}$. When multiplying powers with same base, add the exponents.

98. What is the value of $5^0$?

    1. Any nonzero number raised to the power of 0 equals 1.

99. Simplify: $\text{square root of } 121$.

    11. $\sqrt{121} = 11$ since $11^2 = 121$.

100. What is the cube root of 27?

     3. The number that when cubed equals 27: $3^3 = 27$.

101. Identify the exponent in $6^4$.

     4. The exponent is the power to which the base is raised.

102. What is the principal square root property for $x \ge 0$, $\sqrt{x^2}$?

     $x$. For non-negative $x$, $\sqrt{x^2} = x$ by definition.

103. State the property of exponents for raising a power to a power.

     $(a^m)^n = a^{m \times n}$. When raising a power to a power, multiply the exponents.

104. Simplify $(9^{\frac{1}{2}})^2$.

     9. Power to a power rule: $(a^m)^n = a^{mn}$, so $\frac{1}{2} \times 2 = 1$.

105. Which condition makes $f(x)=a^x$ an increasing function?

     $a>1$. Base greater than 1 creates exponential growth.

106. What is the simplification identity for all real $x$, $\sqrt{x^2}$?

     $|x|$. Absolute value needed since $x$ can be negative.

107. What is the simplified value of $27^{\frac{2}{3}}$?

     $9$. $27^{2/3} = (3^3)^{2/3} = 3^2 = 9$.

108. Express $64$ as a power of $4$.

     $4^3$. Find what power of 4 gives 64: $4 \times 4 \times 4 = 64$.

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