ACT Math Flashcards: Equivalent Expressions

What this deck covers

This deck focuses on Equivalent Expressions, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

Flashcard 4: Identify the equivalent expression for $x^2 + 2x + 1$.

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

Flashcard 5: What is the zero exponent rule for $a\neq 0$?

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

Flashcard 6: What expression is equivalent to $9(x + 1) - x$?

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

Flashcard 7: What is the quotient of powers rule for the same base?

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

Flashcard 8: What is the difference of squares factoring pattern?

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

Flashcard 9: What is the simplified form of $3(x + 4) - 2(x - 1)$?

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

Flashcard 10: What is the common factoring pattern for $ab+ac$?

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

Flashcard 11: What is the factoring pattern for $a^2+2ab+b^2$?

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

Flashcard 12: Simplify the expression: $3x(x + 4) - 2x(x - 1)$.

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

Flashcard 13: Identify the equivalent expression for $2(x + 3) + 4x$.

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

Flashcard 14: What is the equivalent expression for $x^3 x^5$ using exponent rules?

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

Flashcard 15: What is the difference of squares factoring pattern?

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

Flashcard 16: What is the product of powers rule for the same base?

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

Flashcard 17: Factor $4x^2 - 9$ completely.

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

Flashcard 18: What is the zero property of multiplication written as an equivalent expression?

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

Flashcard 19: Simplify the expression: $5(x - 3) + 2(3 - x)$.

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

Flashcard 20: Simplify the expression: $2x(x + 3) + 3(x + 3)$.

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

Flashcard 21: What is the power of a power rule?

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

Flashcard 22: What property justifies rewriting $a+b$ as $b+a$?

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

Flashcard 23: What is the equivalent expression for $a^2 - 2ab + b^2$?

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

Flashcard 24: Which expression is equivalent to $4x^2 - 9$?

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

Flashcard 25: What is the product of $(x + 3)$ and $(x - 3)$?

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

Flashcard 26: Which expression is equivalent to $3(x - 4) + 2x$?

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

Flashcard 27: What is the simplified form of $3(x + 4) - 2(x - 1)$?

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

Flashcard 28: What is the equivalent form of $x^2 - 2x + 1$?

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

Flashcard 29: What is the equivalent expression for $\frac{3}{x}+ \frac{2}{x}$ for $x\neq 0$?

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

Flashcard 30: Factor the expression $x^2 - 7x + 10$.

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

Flashcard 31: What is the factoring pattern for $a^2+2ab+b^2$?

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

Flashcard 32: What is the factored form of $x^2 - 4$?

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

Flashcard 33: What is the equivalent expression for $3(x + 4)$?

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

Flashcard 34: What property justifies rewriting $(a+b)+c$ as $a+(b+c)$?

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

Flashcard 35: Find the equivalent expression for $3(a + b) - 2a$.

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

Flashcard 36: What is the power of a quotient rule?

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

Flashcard 37: What property justifies rewriting $(ab)c$ as $a(bc)$?

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

Flashcard 38: What is the equivalent expression for $3(x-4)$ after distributing?

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

Flashcard 39: What is the power of a power rule?

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

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

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

Flashcard 41: What is the equivalent expression for $-2(5x-3)$ after distributing?

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

Flashcard 42: Simplify the expression: $3x(x + 4) - 2x(x - 1)$.

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

Flashcard 43: Simplify: $5x - 3(x - 2) + 4$.

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

Flashcard 44: What is the equivalent expression for $\frac{x}{4}+\frac{x}{2}$ after combining terms?

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

Flashcard 45: What is the equivalent expression for $(x+3)^2$ after expanding?

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

Flashcard 46: Factor the expression $x^2 + 5x + 6$.

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

Flashcard 47: Simplify: $2(x^2 + 3x + 2) - (x^2 + 1)$.

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

Flashcard 48: What property justifies rewriting $ab$ as $ba$?

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

Flashcard 49: What is the simplified form of $4x - 2(3x - 5)$?

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

Flashcard 50: What is the equivalent form of $x^2 - 6x + 9$?

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

Flashcard 51: Simplify: $4(x + 3) - 2x$.

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

Flashcard 52: What is the negative exponent rule for $a\neq 0$?

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

Flashcard 53: Express $6a + 9b$ as a product of factors.

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

Flashcard 54: What is the identity property for addition written as an equivalent expression?

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

Flashcard 55: What is the factored form of $x^2 + 5x + 6$?

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

Flashcard 56: Find the equivalent expression for $x^2 - 10x + 25$.

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

Flashcard 57: What is the power of a product rule?

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

Flashcard 58: Simplify: $x^2 + 6x + 9 - (x + 3)^2$.

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

Flashcard 59: Factor the expression $x^2 + 5x + 6$.

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

Flashcard 60: What is the equivalent expression for $(x-5)(x+5)$ after multiplying?

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

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

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

Flashcard 62: What is the equivalent expression for $\frac{2x+4}{2}$?

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

Flashcard 63: What is the power of a product rule?

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

Flashcard 64: What is the product of powers rule for the same base?

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

Flashcard 65: What is the power of a quotient rule?

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

Flashcard 66: Factor $4x^2 - 9$ completely.

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

Flashcard 67: Identify the equivalent expression for $x^2 + 2x + 1$.

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

Flashcard 68: Identify the equivalent expression for $6x - 2(x + 3)$.

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

Flashcard 69: Simplify: $4(x + 3) - 2x$.

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

Flashcard 70: Simplify the expression: $3xy + 4x - 2xy - x$.

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

Flashcard 71: What property justifies rewriting $a(b+c)$ as $ab+ac$?

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

Flashcard 72: What is the equivalent expression for $5x-3(x+2)$ after simplifying?

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

Flashcard 73: Which expression is equivalent to $x^2 + 4x + 4$?

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

Flashcard 74: Identify the equivalent expression for $4(x + 2) + 3x$.

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

Flashcard 75: What is the simplified form of $8x - 3x + 5$?

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

Flashcard 76: What is the equivalent expression for $\frac{6x^2}{3x}$ for $x \neq 0$?

Answer: $2x$. Divide coefficients and subtract exponents.

Flashcard 77: Find the equivalent expression for $x^2 + 2xy + y^2$.

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

Flashcard 78: Simplify: $x^2 + 6x + 9 - (x + 3)^2$.

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

Flashcard 79: What is the square of a binomial formula for $(a+b)^2$?

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

Flashcard 80: Simplify: $x(x + 5) - 2(x + 5)$.

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

Flashcard 81: Simplify the expression: $4x - 2(x + 3)$.

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

Flashcard 82: What is the factoring pattern for $a^2-2ab+b^2$?

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

Flashcard 83: What is the product of $(x + 3)$ and $(x - 3)$?

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

Flashcard 84: What is the expanded form of $(x + 2)^2$?

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

Flashcard 85: What is the common factoring pattern for $ab+ac$?

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

Flashcard 86: What is the equivalent expression for $a^2 - 2ab + b^2$?

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

Flashcard 87: Rewrite $12x + 8$ as a product of its factors.

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

Flashcard 88: Express $6a + 9b$ as a product of factors.

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

Flashcard 89: Simplify the expression: $2(a + b) + 3(a + b)$.

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

Flashcard 90: What is the equivalent form of $x^2 - 25$?

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

Flashcard 91: Identify the equivalent expression for $3x(x - 2) + 4x$.

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

Flashcard 92: What is an equivalent expression for $4a^2 - 9b^2$?

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

Flashcard 93: What is the equivalent expression for $x^2 - 4x + 4$?

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

Flashcard 94: Rewrite $5(x - 1) + 3(x + 2)$ in simplest form.

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

Flashcard 95: What is the quotient of powers rule for the same base?

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

Flashcard 96: Which expression is equivalent to $x^2 + 4x + 4$?

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

Flashcard 97: Which expression is equivalent to $3(x - 4) + 2x$?

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

Flashcard 98: What is the equivalent expression for $3(x + 4)$?

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

Flashcard 99: What is the simplified form of $4x - 2(3x - 5)$?

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

Flashcard 100: What is the equivalent expression for $a^2 - 4$?

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