Equivalent Expressions - PSAT Math

1. What is the equivalent expression for $(x-3)^2$ expanded?

   $x^2-6x+9$. $(x-3)(x-3) = x^2 - 3x - 3x + 9 = x^2 - 6x + 9$.

2. What is the equivalent expression for $\frac{6x^2y}{3xy^2}$ after simplifying?

   $\frac{2x}{y}$. Cancel common factors: $\frac{6}{3} = 2$, $\frac{x^2}{x} = x$, $\frac{y}{y^2} = \frac{1}{y}$.

3. What is the factored equivalent expression for $x^2-9$?

   $(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.

4. What is the power of a product rule for $\left(ab\right)^n$?

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

5. What is the exponent rule for a power of a power: $(a^m)^n$?

   $a^{mn}$. When raising a power to a power, multiply the exponents.

6. What is the simplified equivalent expression for $\left(2x^3y\right)^2$?

   $4x^6y^2$. Apply power to each factor: $(2)^2(x^3)^2(y)^2=4x^6y^2$.

7. What is the factored form of $6x+9$ using the greatest common factor?

   $3(2x+3)$. Factor out GCF of 3: $6x + 9 = 3(2x) + 3(3)$.

8. What is the equivalent expression for $(x+6)(x-6)$ after expanding?

   $x^2-36$. Difference of squares: $(a+b)(a-b) = a^2-b^2$.

9. What property justifies rewriting $ab$ as $ba$?

   Commutative property of multiplication. Order of multiplication doesn't affect the product.

10. What is the equivalent expression for $x^0$ for $x\ne 0$?

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

11. What is the equivalent expression for $x^2+5x$ factored completely?

    $x(x+5)$. Factor out common $x$ from both terms.

12. What is the factored equivalent expression for $6x+9$?

    $3(2x+3)$. Factor out the GCF of $3$ from both terms.

13. What property justifies rewriting $ab+ac$ as $a(b+c)$?

    Factoring using the distributive property. Pulling out the common factor from a sum.

14. What does it mean for two algebraic expressions to be equivalent?

    They have the same value for all values of the variables. Equivalent expressions produce identical outputs for any input values.

15. What property justifies rewriting $a+b$ as $b+a$?

    Commutative property of addition. Order of addition doesn't affect the sum.

16. What is the equivalent expression for $0.4x$ written as a fraction coefficient?

    $\frac{2}{5}x$. Convert decimal to fraction: $0.4 = \frac{2}{5}$.

17. What is an equivalent expression to $3(x+4)$?

    $3x+12$. Distribute $3$ to both $x$ and $4$.

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

    $a^{-n}=\frac{1}{a^n}$. A negative exponent means reciprocal with positive exponent.

19. What is the simplified equivalent expression for $3(x-4)+2x$?

    $5x-12$. Distribute $3$, then combine like terms: $3x-12+2x=5x-12$.

20. What distributive property identity rewrites $ab+ac$ as a product?

    $ab+ac=a(b+c)$. Factor out the common term $a$ from both parts.

21. What is the exponent rule for multiplying same-base powers: $a^m\cdot a^n$?

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

22. What is the equivalent expression for $7x+14$ factored completely?

    $7(x+2)$. Factor out GCF of 7 from both terms.

23. What is an equivalent expression to $(2x^3)^2$?

    $4x^6$. Apply power rule: $(2)^2=4$ and $(x^3)^2=x^6$.

24. What is the exponent rule for dividing same bases: $\frac{a^m}{a^n}$?

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

25. What is the difference of squares identity for $a^2-b^2$?

    $(a-b)(a+b)$. The difference of squares factors into conjugate binomials.

26. What property justifies rewriting $(a+b)+c$ as $a+(b+c)$?

    Associative property of addition. Grouping doesn't matter when adding.

27. What is the factored equivalent expression for $x^2+10x+25$?

    $(x+5)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.

28. What property justifies rewriting $ab$ as $ba$?

    Commutative property of multiplication. Order doesn't matter when multiplying.

29. What is the simplified form of $3a-2(a-4)$?

    $a+8$. Distribute $-2$: $3a - 2a + 8 = a + 8$.

30. What is the equivalent expression for $6x+9$ written in factored form?

    $3(2x+3)$. Factor out the GCF of 3 from both terms.

31. What is the simplified equivalent expression for $\frac{12x^2y}{3xy^2}$ for $x\ne 0$ and $y\ne 0$?

    $\frac{4x}{y}$. Cancel common factors: $\frac{12x^2y}{3xy^2}=\frac{4x}{y}$.

32. What is the equivalent expression for $2x+3x^2-x+5$ in standard form?

    $3x^2+x+5$. Combine like terms and arrange by degree.

33. What is the equivalent expression for $(x^2)^3$ using exponent rules?

    $x^6$. Power of a power: multiply exponents $(x^2)^3 = x^{2 \cdot 3}$.

34. What is the equivalent expression for $x^2+9x$ written in factored form?

    $x(x+9)$. Factor out the common $x$ from both terms.

35. What is the equivalent expression for $x^3x^5$ using exponent rules?

    $x^8$. When multiplying same base, add exponents: $x^{3+5}$.

36. What property justifies rewriting $(a+b)+c$ as $a+(b+c)$?

    Associative property of addition. Grouping doesn't affect the sum.

37. What is the equivalent expression for $x^2-16$ written in factored form?

    $(x-4)(x+4)$. Difference of squares factoring: $a^2 - b^2 = (a-b)(a+b)$.

38. What property justifies rewriting $a(b+c)$ as $ab+ac$?

    Distributive property. Multiplication distributes over addition.

39. What is the exponent rule for dividing same-base powers: $\frac{a^m}{a^n}$ for $a\ne 0$?

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

40. What is the simplified equivalent expression for $3(x-4)+2x$?

    $5x-12$. Distribute $3$, then combine like terms: $3x-12+2x=5x-12$.

41. What is the factored form of $6x+9$?

    $3(2x+3)$. Factor out the GCF of 3 from both terms.

42. What is the simplified equivalent expression for $7a-2a+4$?

    $5a+4$. Combine like terms: $7a - 2a = 5a$.

43. What is the equivalent expression for $2x^2+5x+3$ factored completely?

    $(2x+3)(x+1)$. Find factors of 3 that add to 5 when considering 2x.

44. What property justifies rewriting $a+b$ as $b+a$?

    Commutative property of addition. Order doesn't matter when adding.

45. What is the square of a binomial formula for $(a+b)^2$?

    $a^2+2ab+b^2$. Expand using FOIL: first, outer, inner, last terms.

46. What is the identity for factoring out $-1$ from an expression $-(a-b)$?

    $-(a-b)=-a+b$. Distributing $-1$ changes the sign of each term.

47. What is the equivalent expression for $(2x-5)^2$?

    $4x^2-20x+25$. Perfect square: $(a-b)^2 = a^2 - 2ab + b^2$.

48. What is the exponent rule for a quotient to a power: $\left(\frac{a}{b}\right)^n$ for $b\ne 0$?

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

49. What is the simplified form of $\frac{6x^2y}{3xy}$ assuming $x\ne 0$ and $y\ne 0$?

    $2x$. Cancel common factors: $\frac{6x^2y}{3xy}=\frac{6x}{3}=2x$.

50. What property justifies rewriting $(ab)c$ as $a(bc)$?

    Associative property of multiplication. Grouping of multiplication doesn't affect the product.

51. What is the simplified form of $(2x-3)^2$?

    $4x^2-12x+9$. Apply $(a-b)^2$: $(2x)^2-2(2x)(3)+3^2=4x^2-12x+9$.

52. What is the equivalent expression for $\frac{6x}{9}$ in simplest form?

    $\frac{2x}{3}$. Divide numerator and denominator by their GCF of 3.

53. What is the equivalent expression for $5(2x-3)$ after distributing?

    $10x-15$. Multiply $5$ by each term: $5 \cdot 2x + 5 \cdot (-3)$.

54. What is the simplified equivalent expression for $\frac{x^2-4}{x-2}$ for $x\ne 2$?

    $x+2$. Factor and cancel: $\frac{(x-2)(x+2)}{x-2}=x+2$.

55. What identity expands a product of conjugates: $(a+b)(a-b)$?

    $(a+b)(a-b)=a^2-b^2$. Product of conjugates gives difference of squares.

56. What property justifies rewriting $a(b+c)$ as $ab+ac$?

    Distributive property. Multiplying a term by a sum equals multiplying by each addend.

57. What is the factored form of $x^2+6x+9$?

    $(x+3)^2$. Perfect square trinomial: $(x+3)(x+3)=(x+3)^2$.

58. What is the expanded equivalent expression for $(x+7)(x-2)$?

    $x^2+5x-14$. FOIL: $x^2-2x+7x-14=x^2+5x-14$.

59. What is the equivalent expression for $-2(5x+3)$?

    $-10x-6$. Distribute -2: $-2 \cdot 5x + (-2) \cdot 3 = -10x - 6$.

60. What is the power rule for a power: $(a^m)^n$?

    $a^{mn}$. When raising a power to a power, multiply the exponents.

61. What is the equivalent expression for $-4(2x+5)$ after distributing?

    $-8x-20$. Multiply -4 by each term: $-4 \cdot 2x = -8x$ and $-4 \cdot 5 = -20$.

62. What is an equivalent expression to $5x-2(x-3)$?

    $3x+6$. Distribute $-2$, then combine like terms: $5x-2x+6=3x+6$.

63. What is the factored form of $6x+12$ using the greatest common factor?

    $6(x+2)$. GCF of 6 and 12 is 6; factor it out.

64. What is the factored form of $4x^2-12x$?

    $4x(x-3)$. Factor out the GCF of $4x$ from both terms.

65. What is the factored form of $x^2-5x-14$?

    $(x-7)(x+2)$. Find factors of -14 that add to -5: -7 and 2.

66. What is the factored form of $x^2-16$?

    $(x-4)(x+4)$. Difference of squares: $a^2-b^2 = (a-b)(a+b)$.

67. What is the simplified form of $(x+4)(x-4)$?

    $x^2-16$. Apply difference of squares: $(x)^2-(4)^2=x^2-16$.

68. What identity expands a perfect square: $(a+b)^2$?

    $(a+b)^2=a^2+2ab+b^2$. Square the first, twice the product, square the second.

69. What is the result of factoring out the greatest common factor in $6x+9$?

    $3(2x+3)$. GCF of $6x$ and $9$ is $3$; factor it out.

70. What is the expanded equivalent expression for $(x+3)(x+2)$?

    $x^2+5x+6$. FOIL: $x^2+2x+3x+6=x^2+5x+6$.

71. What is the simplified equivalent expression for $-(2x-7)+x$?

    $-x+7$. Distribute negative: $-2x+7+x=-x+7$.

72. What is the equivalent expression for $4x-(2x-7)$ simplified?

    $2x+7$. Distribute negative: $4x - 2x + 7 = 2x + 7$.

73. What is the equivalent expression for $2(x+3)+5(x-1)$ simplified?

    $7x+1$. Distribute both: $2x + 6 + 5x - 5 = 7x + 1$.

74. What is the equivalent expression for $x^2-16$ factored completely?

    $(x-4)(x+4)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

75. What property justifies rewriting $ab$ as $ba$?

    Commutative property of multiplication. Order doesn't matter when multiplying numbers.

76. What is the simplified form of $\frac{1}{2}x+\frac{3}{4}x$?

    $\frac{5}{4}x$. Convert to common denominator: $\frac{2}{4}x+\frac{3}{4}x=\frac{5}{4}x$.

77. What is the equivalent expression for $(x^2)^3$?

    $x^6$. Power rule: multiply exponents when raising power to power.

78. What is an equivalent expression to $7(x-2)+3(x-2)$?

    $10(x-2)$. Factor out common term $(x-2)$: $7+3=10$.

79. What is the equivalent expression for $4x-8$ factored completely?

    $4(x-2)$. Factor out GCF of 4 from both terms.

80. What property justifies rewriting $(ab)c$ as $a(bc)$?

    Associative property of multiplication. Grouping doesn't affect the product.

81. What is the equivalent expression for $(x+3)^2$ after expanding?

    $x^2+6x+9$. Perfect square: $(a+b)^2 = a^2+2ab+b^2$.

82. What is the equivalent expression for $-2(5x+1)$?

    $-10x-2$. Distribute $-2$ to both terms inside parentheses.

83. What is the equivalent expression for $4x^2y-3xy^2+2x^2y$ after combining like terms?

    $6x^2y-3xy^2$. Combine like terms: $4x^2y + 2x^2y = 6x^2y$.

84. What is the simplified form of $\frac{2}{x}+\frac{3}{x}$ for $x\ne 0$?

    $\frac{5}{x}$. Add fractions with same denominator: $\frac{2+3}{x} = \frac{5}{x}$.

85. What is the expansion of $(2x+1)^2$?

    $4x^2+4x+1$. Apply $(a+b)^2 = a^2 + 2ab + b^2$ with $a=2x$, $b=1$.

86. What property justifies rewriting $ab$ as $ba$?

    Commutative property of multiplication. Order doesn't matter when multiplying.

87. What is the equivalent expression for $\frac{3}{x}+\frac{2}{x}$ assuming $x\ne 0$?

    $\frac{5}{x}$. Add fractions with same denominator: $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$.

88. What property justifies rewriting $ab+ac$ as $a(b+c)$?

    Distributive property (factoring). Reverse of distribution; factor out common term.

89. What is the simplified equivalent expression for $5x-3(x+2)$?

    $2x-6$. Distribute $-3$: $5x-3x-6=2x-6$.

90. What property justifies rewriting $ab$ as $ba$?

    Commutative property of multiplication. Order doesn't matter when multiplying.

91. What is the equivalent expression for $(x+3)^2$?

    $x^2+6x+9$. Perfect square: $(a+b)^2 = a^2 + 2ab + b^2$.

92. What is the simplified form of $\frac{6x^2y}{3xy}$?

    $2x$. Cancel common factors: $\frac{6x^2y}{3xy} = \frac{6x}{3} = 2x$.

93. What is the equivalent expression for $x^3\text{ }x^5$?

    $x^8$. Product rule: add exponents when multiplying same base.

94. What property justifies rewriting $a(b+c)$ as $ab+ac$?

    Distributive property. Multiplying a term by a sum equals the sum of the products.

95. What is the simplified equivalent expression for $\frac{x^2-9}{x-3}$ when $x\ne 3$?

    $x+3$. Factor numerator: $\frac{(x-3)(x+3)}{x-3}=x+3$.

96. What is the exponent rule for multiplying like bases: $a^m \times a^n$?

    $a^{m+n}$. When multiplying same bases, add the exponents.

97. What is the equivalent expression for $4x+12$ after factoring out the GCF $4$?

    $4(x+3)$. Factor out the common factor $4$ from both terms.

98. What expression is equivalent to $x^2+5x+6$ when factored completely?

    $(x+2)(x+3)$. Find two numbers that multiply to $6$ and add to $5$: $2$ and $3$.

99. What is the exponent rule for dividing like bases: $\frac{a^m}{a^n}$ for $a\ne 0$?

    $a^{m-n}$. When dividing same bases, subtract the exponents.

100. What is the equivalent expression for $\frac{x^7}{x^3}$ using exponent rules?

     $x^4$. When dividing same base, subtract exponents: $x^{7-3}$.

101. What is the equivalent expression for $\frac{6x^2}{3x}$ for $x\ne 0$?

     $2x$. Cancel common factor of $3x$: $\frac{6x^2}{3x} = \frac{6x \cdot x}{3x} = 2x$.

102. What is the equivalent expression for $(x+7)(x-7)$?

     $x^2-49$. Difference of squares: $(a+b)(a-b) = a^2 - b^2$.

103. What is the equivalent expression for $x^2+10x+25$ factored completely?

     $(x+5)^2$. Perfect square trinomial: $a^2 + 2ab + b^2 = (a+b)^2$.

104. What is the equivalent expression for $9x^2-6x$ factored completely?

     $3x(3x-2)$. Factor out GCF of $3x$ from both terms.

105. What is the distributive property for factoring out a common factor?

     $ab+ac=a(b+c)$. Factor out the common term $a$ from both addends.

106. What is the equivalent expression for $(x-5)^2$ after expanding?

     $x^2-10x+25$. Perfect square: $(a-b)^2 = a^2-2ab+b^2$.

107. What property justifies rewriting $a(b+c)$ as $ab+ac$?

     Distributive property. Multiplication distributes over addition.

108. What is the simplified form of $\frac{1}{2}x+3x$?

     $\frac{7}{2}x$. Convert to common form: $\frac{1}{2}x + \frac{6}{2}x = \frac{7}{2}x$.

109. What is the equivalent expression for $(x+5)(x+2)$ in standard form?

     $x^2+7x+10$. FOIL: $x^2 + 2x + 5x + 10 = x^2 + 7x + 10$.

110. What is the correct expansion of $-2(5x+1)$?

     $-10x-2$. Distribute: $-2 \cdot 5x = -10x$ and $-2 \cdot 1 = -2$.

111. What is the simplified equivalent expression for $\frac{2}{x}+\frac{3}{x}$ when $x\ne 0$?

     $\frac{5}{x}$. Same denominator: $\frac{2+3}{x}=\frac{5}{x}$.

112. What is the factored equivalent expression for $x^2-9$?

     $(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.

113. What expression is equivalent to $2(x-5)+3x$ after simplifying?

     $5x-10$. Distribute $2$, then combine like terms: $2x-10+3x=5x-10$.

114. What does it mean for two expressions to be equivalent?

     They have the same value for all allowed values of the variables. Equivalent expressions produce identical outputs for any valid input.

115. What is the equivalent expression for $\frac{x}{3}+\frac{2x}{3}$?

     $x$. Add fractions with same denominator: $\frac{x + 2x}{3} = \frac{3x}{3}$.

116. What is the equivalent expression for $\frac{2}{3}x+\frac{1}{6}x$?

     $\frac{5}{6}x$. Find LCD of 6, then add: $\frac{4}{6}x + \frac{1}{6}x = \frac{5}{6}x$.

117. What is the equivalent expression for $(x+4)(x-4)$ in expanded form?

     $x^2-16$. Difference of squares pattern: $(a+b)(a-b) = a^2 - b^2$.

118. What is the simplified equivalent expression for $2(3x+5)-4x$?

     $2x+10$. Distribute $2$, then combine: $6x+10-4x=2x+10$.

119. What is the equivalent expression for $\frac{2x}{4}$ in simplest form?

     $\frac{x}{2}$. Simplify by dividing numerator and denominator by 2.

120. What is the equivalent expression for $3(x-4)$?

     $3x-12$. Distribute 3 to both terms: $3 \cdot x - 3 \cdot 4$.

121. What property justifies rewriting $ab+ac$ as $a(b+c)$?

     Distributive property (factoring out $a$). Reverse distribution pulls out the common factor.

122. What is the equivalent expression for $x^2-16$ factored over the integers?

     $(x-4)(x+4)$. Difference of squares: $a^2 - b^2 = (a - b)(a + b)$.

123. What is the product of $(x+3)(x+2)$ written as a trinomial?

     $x^2+5x+6$. Use FOIL: $x^2 + 2x + 3x + 6 = x^2 + 5x + 6$.

124. What is the simplified form of $3(x+2)-2(x-5)$?

     $x+16$. Distribute both, then combine: $3x+6-2x+10=x+16$.

125. What is an equivalent expression to $(x+5)^2$?

     $x^2+10x+25$. Use $(a+b)^2=a^2+2ab+b^2$ with $a=x$ and $b=5$.

126. What is the factored equivalent expression for $x^2-16$?

     $(x-4)(x+4)$. Difference of squares: $16=4^2$.

127. What is an equivalent expression to $2x+7x-4$?

     $9x-4$. Combine like terms: $2x+7x=9x$.

128. What is the equivalent expression for $7(x-3)$ after distributing?

     $7x-21$. Multiply 7 by each term: $7 \cdot x = 7x$ and $7 \cdot (-3) = -21$.

129. Which expression is equivalent to $2(x+4)+3(x-1)$?

     $5x+5$. Distribute: $2x + 8 + 3x - 3 = 5x + 5$.

130. What is the equivalent expression for $\frac{2}{x}+\frac{3}{x}$ after combining like terms?

     $\frac{5}{x}$. Same denominator, so add numerators: $2+3 = 5$.

131. What is the power of a quotient rule for $\left(\frac{a}{b}\right)^n$ with $b\ne 0$?

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

132. What is the equivalent expression for $-2(5-x)$?

     $2x-10$. Distribute -2: $-2 \cdot 5 + (-2)(-x) = -10 + 2x$.

133. What is the factored form of $x^2-9$?

     $(x-3)(x+3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

134. What property justifies rewriting $ab+ac$ as $a(b+c)$?

     Factoring using the distributive property. Reverse of distributive property; factor out common term.

135. What is the exponent rule for a product to a power: $(ab)^n$?

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

136. What is the simplified equivalent expression for $\left(2x^2y\right)^2$?

     $4x^4y^2$. Apply power rule: $2^2 \cdot (x^2)^2 \cdot y^2 = 4x^4y^2$.

137. What is the expansion of $(x-5)^2$?

     $x^2-10x+25$. Apply $(a-b)^2 = a^2 - 2ab + b^2$ with $a=x$, $b=5$.

138. What property justifies rewriting $(a+b)+c$ as $a+(b+c)$?

     Associative property of addition. Grouping doesn't affect the sum.

139. What is the equivalent expression for $\frac{x^2}{x}$ assuming $x\ne 0$?

     $x$. Cancel common factor $x$ from numerator and denominator.

140. What is the simplified form of $\frac{x^2-9}{x-3}$, assuming $x\ne 3$?

     $x+3$. Factor numerator: $\frac{(x-3)(x+3)}{x-3} = x+3$.

141. What is the equivalent expression for $2x+5x$?

     $7x$. Combine like terms: $2x + 5x = (2 + 5)x = 7x$.

142. What property justifies rewriting $a+b$ as $b+a$?

     Commutative property of addition. Order doesn't matter when adding numbers.

143. What is the equivalent expression for $\frac{5}{10}$ in simplest form?

     $\frac{1}{2}$. Divide numerator and denominator by GCF of 5.

144. What is the equivalent expression for $(x+7)+3$ written without parentheses?

     $x+10$. Associative property allows removing parentheses: $x + 7 + 3$.

145. What property justifies rewriting $ab+ac$ as $a(b+c)$?

     Distributive property (factoring). Factoring reverses distribution by pulling out common factors.

146. What is the simplified form of $7-(2x-5)$?

     $12-2x$. Distribute the negative: $7-2x+5=12-2x$.

147. What is the distributive property for subtraction, using variables $a$, $b$, and $c$?

     $a(b-c)=ab-ac$. Distribute the factor and keep the subtraction sign.

148. What is the simplified equivalent expression for $\frac{5}{x}+\frac{3}{x}$ for $x\ne 0$?

     $\frac{8}{x}$. Same denominator, so add numerators: $\frac{5+3}{x}=\frac{8}{x}$.

149. What is the equivalent expression for $(2x+3)^2$ in expanded form?

     $4x^2+12x+9$. Square of binomial: $(a+b)^2 = a^2 + 2ab + b^2$.

150. What is the correct expansion of $3(x-4)$?

     $3x-12$. Distribute: $3 \cdot x = 3x$ and $3 \cdot (-4) = -12$.

151. What is the definition of like terms?

     Terms with the same variable part and exponents. Like terms can be combined because they have identical variables.

152. What property justifies rewriting $(a+b)+c$ as $a+(b+c)$?

     Associative property of addition. Grouping of addition doesn't affect the sum.

153. What is the equivalent expression for $-2(x-7)$ after distributing?

     $-2x+14$. Distribute $-2$: $-2 \cdot x + (-2) \cdot (-7) = -2x + 14$.

154. What is the simplified form of $x^2+3x+2x+6$?

     $x^2+5x+6$. Combine like terms: $3x + 2x = 5x$.

155. What is the distributive property written with variables?

     $a(b+c)=ab+ac$. Multiply $a$ by each term inside the parentheses.

156. What is the definition of a factor of an expression?

     An expression that divides it with no remainder. A factor multiplies to give the original expression.

157. What is the expanded equivalent expression for $(x+4)^2$?

     $x^2+8x+16$. Perfect square: $(a+b)^2=a^2+2ab+b^2$.

158. What is the expanded equivalent expression for $(x-5)^2$?

     $x^2-10x+25$. Perfect square: $(a-b)^2=a^2-2ab+b^2$.

159. What is the equivalent expression for $6x-9$ after factoring out the GCF $3$?

     $3(2x-3)$. Factor out the common factor $3$ from both terms.

160. What is the exponent rule for multiplying same bases: $a^m \cdot a^n$?

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

161. What is an equivalent factored form of $x^2-16$?

     $(x-4)(x+4)$. Recognize $16=4^2$, so $x^2-16=x^2-4^2$ is a difference of squares.

162. Which expression is equivalent to $\frac{1}{2}(8x+6)$?

     $4x+3$. Distribute $\frac{1}{2}$: $\frac{1}{2}\cdot^8x=4x$ and $\frac{1}{2}\cdot^6=3$.

163. What is the equivalent expression for $4x^2-9x^2$?

     $-5x^2$. Combine like terms: $4x^2 - 9x^2 = (4 - 9)x^2 = -5x^2$.

164. What is the simplified equivalent expression for $2x+3x-5$?

     $5x-5$. Combine like terms: $2x + 3x = 5x$.

165. What is the simplified form of $2x+5x-3$?

     $7x-3$. Combine like terms: $2x + 5x = 7x$.

166. What property justifies rewriting $a+b$ as $b+a$?

     Commutative property of addition. Order doesn't matter when adding.

167. What is the definition of a term in an algebraic expression?

     A part separated by addition or subtraction. Terms are the individual parts combined by $+$ or $-$.

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

     $a^0=1$. Any nonzero number raised to the zero power equals one.

169. What is the equivalent expression for $x^2-9$ factored completely?

     $(x-3)(x+3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

170. What is the equivalent expression for $(x+2)(x+5)$?

     $x^2+7x+10$. FOIL: First, Outer, Inner, Last terms multiplied.

171. What is the simplified form of $5a-3(2a-4)$?

     $-a+12$. Distribute $-3$, then combine: $5a-6a+12=-a+12$.

172. What is the simplified form of $\frac{x^2-9}{x-3}$ assuming $x\ne 3$?

     $x+3$. Factor numerator: $\frac{(x-3)(x+3)}{x-3}=x+3$.

173. What is the factored form of $x^2+7x+12$?

     $(x+3)(x+4)$. Find factors of 12 that add to 7: 3 and 4.

174. What is the equivalent expression for $3(x+4)$ after distributing?

     $3x+12$. Multiply $3$ by each term: $3 \cdot x + 3 \cdot 4$.

175. What property justifies rewriting $(ab)c$ as $a(bc)$?

     Associative property of multiplication. Grouping doesn't affect the product.

176. What property justifies rewriting $a(b+c)$ as $ab+ac$?

     Distributive property. Multiplication distributes over addition.

177. What is the equivalent expression for $x^{-3}$ written with a positive exponent?

     $\frac{1}{x^3}$. Negative exponent means reciprocal: $x^{-3} = \frac{1}{x^3}$.

178. What identity expands a perfect square: $(a-b)^2$?

     $(a-b)^2=a^2-2ab+b^2$. Same pattern but subtract twice the product.

179. What is the equivalent expression for $\frac{x^6}{x^2}$ for $x\ne 0$?

     $x^4$. Quotient rule: subtract exponents when dividing same base.

180. What expression is equivalent to $x^2-9$ when factored completely?

     $(x-3)(x+3)$. Difference of squares: $x^2-9=x^2-3^2=(x-3)(x+3)$.

181. What is the simplified equivalent expression for $2(x+5)-3(x-1)$?

     $-x+13$. Distribute both, then combine: $2x+10-3x+3=-x+13$.

182. What property justifies rewriting $(ab)c$ as $a(bc)$?

     Associative property of multiplication. Grouping doesn't affect the product.

183. What is an equivalent expression to $4a^2\cdot 3a^5$?

     $12a^7$. Multiply coefficients and add exponents: $4\cdot^3=12$, $a^{2+5}=a^7$.

184. What is the equivalent expression for $4x-2(x+5)$?

     $2x-10$. Distribute $-2$, then combine like terms: $4x - 2x - 10$.

185. What is the power of a power rule for $\left(a^m\right)^n$?

     $a^{mn}$. When raising a power to a power, multiply the exponents.

186. What is the simplified equivalent expression for $\frac{3x}{6}$ in lowest terms?

     $\frac{x}{2}$. Simplify by dividing both numerator and denominator by $3$.

187. What is the equivalent expression for $(x+2)(x+5)$ in expanded form?

     $x^2+7x+10$. FOIL: First, Outer, Inner, Last terms multiply.

188. What is the equivalent expression for $x^2+6x+9$ factored completely?

     $(x+3)^2$. Perfect square trinomial: $a^2 + 2ab + b^2 = (a+b)^2$.

189. What is the simplified form of $4x-2(x+3)$?

     $2x-6$. Distribute $-2$, then combine: $4x-2x-6=2x-6$.

190. What is an equivalent expression to $\frac{6x^4}{2x}$?

     $3x^3$. Divide coefficients and subtract exponents: $\frac{6}{2}=3$, $x^{4-1}=x^3$.

191. What property justifies rewriting $a+b$ as $b+a$?

     Commutative property of addition. Order doesn't matter when adding.

192. What is the distributive property written with variables $a$, $b$, and $c$?

     $a(b+c)=ab+ac$. Multiply $a$ by each term inside the parentheses.

193. What is the equivalent expression for $3x+2x^2-x+5$ in standard form?

     $2x^2+2x+5$. Combine like terms and order by decreasing degree.

194. What is the equivalent expression for $\frac{1}{3}(6x+9)$?

     $2x+3$. Distribute $\frac{1}{3}$: $\frac{1}{3} \cdot 6x + \frac{1}{3} \cdot 9$.

195. What distributive property identity rewrites $a(b+c)$ as a sum?

     $a(b+c)=ab+ac$. Multiply $a$ by each term inside the parentheses.

196. What property justifies rewriting $(ab)c$ as $a(bc)$?

     Associative property of multiplication. Grouping doesn't matter when multiplying.

197. What is the equivalent expression for $3x-2(x-5)$ after simplifying?

     $x+10$. Distribute -2, then combine: $3x-2x+10 = x+10$.

198. What is the simplified equivalent expression for $\frac{x^5}{x^2}$ for $x\ne 0$?

     $x^3$. Apply quotient rule: $x^{5-2} = x^3$.

199. What property justifies rewriting $(a+b)+c$ as $a+(b+c)$?

     Associative property of addition. Grouping doesn't affect the sum.

200. What is the simplified equivalent expression for $\frac{6x^2}{3x}$ when $x\ne 0$?

     $2x$. Cancel common factor: $\frac{6x^2}{3x}=\frac{6x}{3}=2x$.

201. What is the equivalent expression for $\frac{x^2-9}{x-3}$ for $x\ne 3$?

     $x+3$. Factor numerator: $\frac{(x-3)(x+3)}{x-3} = x+3$.

202. What is the factored form of $x^2+9x$?

     $x(x+9)$. Factor out common $x$ from both terms.

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