Equivalent Expressions - ACT Math

1. What is the equivalent expression for $\frac{y^7}{y^2}$ for $y \neq 0$?

   $y^5$. Subtract exponents when dividing same base: $7-2=5$.

2. What expression is equivalent to $9(x + 1) - x$?

   $8x + 9$. Distribute 9 and combine like terms.

3. What is the zero exponent rule for $a\neq 0$?

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

4. What is the difference of squares factoring pattern?

   $a^2-b^2=(a-b)(a+b)$. Pattern: $(a-b)(a+b)$ gives difference of squares.

5. What is the factoring pattern for $a^2+2ab+b^2$?

   $(a+b)^2$. Perfect square trinomial with positive middle term.

6. Simplify the expression: $3x(x + 4) - 2x(x - 1)$.

   $x^2 + 14x$. Distribute and combine like terms for each degree.

7. Identify the equivalent expression for $2(x + 3) + 4x$.

   $6x + 6$. Distribute 2, then combine like terms: $2x + 6 + 4x$.

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

   $x^8$. Add exponents when multiplying same base: $3+5=8$.

9. What is the product of powers rule for the same base?

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

10. Factor $4x^2 - 9$ completely.

    $(2x - 3)(2x + 3)$. Difference of squares pattern with $a = 2x$ and $b = 3$.

11. Simplify the expression: $5(x - 3) + 2(3 - x)$.

    $3x - 9$. Distribute both terms and combine like terms carefully.

12. Simplify the expression: $2x(x + 3) + 3(x + 3)$.

    $(2x + 3)(x + 3)$. Factor out the common binomial factor $(x + 3)$.

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

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

14. What is the equivalent expression for $a^2 - 2ab + b^2$?

    $(a - b)^2$. Perfect square trinomial pattern with $a$ and $b$.

15. What is the product of $(x + 3)$ and $(x - 3)$?

    $x^2 - 9$. Difference of squares pattern: $(a+b)(a-b) = a^2 - b^2$.

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

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

17. What is the simplified form of $3(x + 4) - 2(x - 1)$?

    $x + 14$. Distribute and combine: $3x + 12 - 2x + 2$.

18. What is the equivalent form of $x^2 - 2x + 1$?

    $(x - 1)^2$. Perfect square trinomial with pattern $a^2 - 2ab + b^2 = (a - b)^2$.

19. What is the equivalent expression for $\frac{3}{x}+ \frac{2}{x}$ for $x\neq 0$?

    $\frac{5}{x}$. Add fractions with same denominator by adding numerators.

20. Factor the expression $x^2 - 7x + 10$.

    $(x - 5)(x - 2)$. Find two numbers that multiply to 10 and add to -7.

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

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

22. What is the equivalent expression for $3(x + 4)$?

    $3x + 12$. Distribute 3 to each term inside the parentheses.

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

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

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

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

25. What is the power of a power rule?

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

26. Simplify: $5x - 3(x - 2) + 4$.

    $2x + 10$. Distribute -3 and combine all like terms.

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

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

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

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

29. Simplify: $4(x + 3) - 2x$.

    $2x + 12$. Distribute 4 and combine like terms.

30. Express $6a + 9b$ as a product of factors.

    $3(2a + 3b)$. Factor out the greatest common factor 3.

31. What is the identity property for addition written as an equivalent expression?

    $a+0=a$. Adding zero leaves the value unchanged.

32. What is the factored form of $x^2 + 5x + 6$?

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

33. What is the power of a product rule?

    $(ab)^n=a^n b^n$. Distribute the exponent to each factor in the product.

34. Factor the expression $x^2 + 5x + 6$.

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

35. What is the equivalent expression for $\frac{2x+4}{2}$?

    $x+2$. Factor out $2$ from numerator and simplify.

36. What is the power of a quotient rule?

    $\big(\frac{a}{b}\big)^n=\frac{a^n}{b^n}$, $b\neq 0$. Distribute the exponent to numerator and denominator.

37. Identify the equivalent expression for $x^2 + 2x + 1$.

    $(x + 1)^2$. Perfect square trinomial with $a=x$ and $b=1$.

38. Identify the equivalent expression for $6x - 2(x + 3)$.

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

39. Simplify the expression: $3xy + 4x - 2xy - x$.

    $xy + 3x$. Combine like terms $xy$ and $x$ separately.

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

    $2x-6$. Distribute first, then combine like terms.

41. Which expression is equivalent to $x^2 + 4x + 4$?

    $(x + 2)^2$. Perfect square trinomial with $a=x$ and $b=2$.

42. Identify the equivalent expression for $4(x + 2) + 3x$.

    $7x + 8$. Distribute 4, then combine like terms: $4x + 8 + 3x$.

43. What is the simplified form of $8x - 3x + 5$?

    $5x + 5$. Combine like terms: $8x - 3x = 5x$.

44. Find the equivalent expression for $x^2 + 2xy + y^2$.

    $(x + y)^2$. Perfect square trinomial with variables $x$ and $y$.

45. Simplify: $x^2 + 6x + 9 - (x + 3)^2$.

    $0$. Both expressions are identical, so their difference is zero.

46. Simplify: $x(x + 5) - 2(x + 5)$.

    $(x - 2)(x + 5)$. Factor out the common binomial factor $(x + 5)$.

47. What is the factoring pattern for $a^2-2ab+b^2$?

    $(a-b)^2$. Perfect square trinomial with negative middle term.

48. What is the common factoring pattern for $ab+ac$?

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

49. Rewrite $12x + 8$ as a product of its factors.

    $4(3x + 2)$. Factor out the greatest common factor 4.

50. Simplify the expression: $2(a + b) + 3(a + b)$.

    $5(a + b)$. Factor out the common term $(a + b)$.

51. What is the equivalent form of $x^2 - 25$?

    $(x - 5)(x + 5)$. Difference of squares pattern: $a^2 - b^2 = (a - b)(a + b)$.

52. Identify the equivalent expression for $3x(x - 2) + 4x$.

    $3x^2 - 2x$. Distribute $3x$, then combine: $3x^2 - 6x + 4x$.

53. What is an equivalent expression for $4a^2 - 9b^2$?

    $(2a - 3b)(2a + 3b)$. Difference of squares pattern with $a = 2a$ and $b = 3b$.

54. What is the quotient of powers rule for the same base?

    $\frac{a^m}{a^n}=a^{m-n}$, $a\neq 0$. When dividing powers with same base, subtract exponents.

55. What is the equivalent expression for $a^2 - 4$?

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

56. What is the zero property of multiplication written as an equivalent expression?

    $a \times 0 = 0$. Any number multiplied by zero equals zero.

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

    $2x$. Divide coefficients and subtract exponents.

58. What is the equivalent expression for $(x-5)(x+5)$ after multiplying?

    $x^2-25$. Difference of squares pattern: $(a-b)(a+b)=a^2-b^2$.

59. Simplify: $4(x^2 - 1) + 2(x + 1)$.

    $4x^2 + 2x - 2$. Distribute both terms and combine like terms.

60. What is the equivalent form of $x^2 - 6x + 9$?

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

61. Which expression is equivalent to $2(x - 5) - x$?

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

62. Find the equivalent expression for $x^2 - 8x + 16$.

    $(x - 4)^2$. Perfect square trinomial: $(x-4)^2 = x^2 - 8x + 16$.

63. Which expression is equivalent to $x^2 - 2x + 1$?

    $(x - 1)^2$. Perfect square trinomial with $a=x$ and $b=1$.

64. Which expression is equivalent to $a^2 - 2a + 1$?

    $(a - 1)^2$. Perfect square trinomial with pattern $a^2 - 2ab + b^2 = (a - b)^2$.

65. What is the equivalent expression for $(x+7)-(2x-5)$ after simplifying?

    $-x+12$. Remove parentheses and combine like terms.

66. What is the identity property for multiplication written as an equivalent expression?

    $a1=a$. Multiplying by one leaves the value unchanged.

67. What is the simplified form of $5x - (2x - 4)$?

    $3x + 4$. Distribute negative sign, then combine: $5x - 2x + 4$.

68. What is an equivalent expression for $2x^2 + 8x$?

    $2x(x + 4)$. Factor out the greatest common factor $2x$.

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

    $2x$. Divide coefficients and subtract exponents: $\frac{6}{3} \cdot \frac{x^2}{x}$.

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

    Distributive property (factoring). Reverse of distribution, pulling common factor outside.

71. What is the simplified form of $x^2 + 2x + 1 - (x + 1)^2$?

    $0$. Both expressions are identical, so their difference is zero.

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

    $-10x+6$. Distribute $-2$ to both $5x$ and $-3$.

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

    Distributive property. Multiplies term outside parentheses by each term inside.

74. What is the definition of equivalence for expressions?

    Same value for all allowed variable values. Two expressions that always produce identical results.

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

    $x+3$. Factor numerator as $(x-3)(x+3)$ and cancel common factor.

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

    $(a-b)^2=a^2-2ab+b^2$. FOIL method with subtraction in second term.

77. What is the equivalent expression for $\frac{3}{x}+ \frac{2}{x}$ for $x\neq 0$?

    $\frac{5}{x}$. Add fractions with same denominator by adding numerators.

78. Which expression is equivalent to $5x + 10$?

    $5(x + 2)$. Factor out the common factor 5.

79. Simplify the expression: $4x - 2(x + 3)$.

    $2x - 6$. Distribute -2 and combine like terms.

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

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

81. Simplify: $3(x^2 - 4) + 2(x^2 + 2)$.

    $5x^2 - 8$. Distribute both terms and combine like terms.

82. What is the factored form of $x^2 - 5x + 6$?

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

83. Which expression is equivalent to $4x^2 - 9$?

    $(2x - 3)(2x + 3)$. Difference of squares: $4x^2 - 9 = (2x)^2 - 3^2$.

84. What is an equivalent expression for $9a^2 - 1$?

    $(3a - 1)(3a + 1)$. Difference of squares pattern with $a = 3a$ and $b = 1$.

85. What is the negative exponent rule for $a\neq 0$?

    $a^{-n}=\frac{1}{a^n}$. Negative exponent means reciprocal of positive exponent.

86. Simplify: $2(x^2 + 3x + 2) - (x^2 + 1)$.

    $x^2 + 6x + 3$. Distribute 2 and -1, then combine like terms.

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

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

88. What is the equivalent expression for $4x+9-2x+1$ after combining like terms?

    $2x+10$. Group and add like terms: $4x-2x=2x$ and $9+1=10$.

89. Rewrite $5(x - 1) + 3(x + 2)$ in simplest form.

    $8x + 1$. Distribute both terms and combine like terms.

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

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

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

    $x+3$. Factor numerator as $(x-3)(x+3)$ and cancel common factor.

92. Find the equivalent expression for $x^2 - 6x + 9$.

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

93. What is the equivalent expression for $\frac{x}{4}+\frac{x}{2}$ after combining terms?

    $\frac{3x}{4}$. Convert to common denominator $4$ before adding.

94. Rewrite $15x - 10$ as a product of factors.

    $5(3x - 2)$. Factor out the greatest common factor 5.

95. What is the expanded form of $(x + 2)^2$?

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

96. Identify the equivalent form of $x^2 + 8x + 16$.

    $(x + 4)^2$. Perfect square trinomial with pattern $a^2 + 2ab + b^2 = (a + b)^2$.

97. Which expression is equivalent to $9 - x^2$?

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

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

    $(a+b)^2=a^2+2ab+b^2$. FOIL method: first, outer, inner, last terms.

99. Rewrite $18x - 12$ as a product of its factors.

    $6(3x - 2)$. Factor out the greatest common factor 6.

100. What is the factored form of $x^2 + 7x + 10$?

     $(x + 2)(x + 5)$. Find two numbers that multiply to 10 and add to 7.

101. What is the simplified form of $7x + 3 - 4x$?

     $3x + 3$. Combine like terms: $7x - 4x = 3x$.

102. What is the equivalent expression for $x^2 - 4x + 4$?

     $(x - 2)^2$. Perfect square trinomial with pattern $a^2 - 2ab + b^2 = (a - b)^2$.

103. Factor the expression $x^2 + 2x - 8$.

     $(x + 4)(x - 2)$. Find two numbers that multiply to -8 and add to 2.

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

     $4x^6$. Square both coefficient and variable: $(2)^2=4$, $(x^3)^2=x^6$.

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

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

106. What is the definition of a negative written as an equivalent expression?

     $-a=(-1)a$. Negative sign is equivalent to multiplying by $-1$.

107. Find the equivalent expression for $x^2 - 10x + 25$.

     $(x - 5)^2$. Perfect square trinomial: $(x-5)^2 = x^2 - 10x + 25$.

108. Find the equivalent expression for $3(a + b) - 2a$.

     $a + 3b$. Distribute 3, then combine like terms: $3a + 3b - 2a$.

109. What is the equivalent expression for $7(x - 2) - 2(x - 3)$?

     $5x - 8$. Distribute both terms and combine like terms.

Study these flashcards