PSAT 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 PSAT 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 $(x-3)^2$ expanded?

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

Flashcard 2: What is the equivalent expression for rac{6x^2y}{3xy^2} after simplifying?

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 9: What property justifies rewriting $ab$ as $ba$?

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

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

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

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

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

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

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

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

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

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

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

Flashcard 15: What property justifies rewriting $a+b$ as $b+a$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 28: What property justifies rewriting $ab$ as $ba$?

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 35: What is the equivalent expression for x^3x^5 using exponent rules?

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

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

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

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

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

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

Answer: Distributive property. Multiplication distributes over addition.

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

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

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

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

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

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

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

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

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

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

Flashcard 44: What property justifies rewriting $a+b$ as $b+a$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 52: What is the equivalent expression for rac{6x}{9} in simplest form?

Answer: rac{2x}{3}. Divide numerator and denominator by their GCF of 3.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 75: What property justifies rewriting $ab$ as $ba$?

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

Flashcard 76: What is the simplified form of rac{1}{2}x+rac{3}{4}x?

Answer: rac{5}{4}x. Convert to common denominator: rac{2}{4}x+rac{3}{4}x=rac{5}{4}x.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 84: What is the simplified form of rac{2}{x}+rac{3}{x} for $x\ne 0$?

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

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

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

Flashcard 86: What property justifies rewriting $ab$ as $ba$?

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

Flashcard 87: What is the equivalent expression for rac{3}{x}+rac{2}{x} assuming $x\ne 0$?

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

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

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

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

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

Flashcard 90: What property justifies rewriting $ab$ as $ba$?

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

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

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

Flashcard 92: What is the simplified form of rac{6x^2y}{3xy}?

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

Flashcard 93: What is the equivalent expression for $x^3 ext{ }x^5$?

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

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

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

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

Answer: $x+3$. Factor numerator: rac{(x-3)(x+3)}{x-3}=x+3.

Flashcard 96: What is the exponent rule for multiplying like bases: $a^m imes a^n$?

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

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

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

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

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

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

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

Flashcard 100: What is the equivalent expression for rac{x^7}{x^3} using exponent rules?

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