GRE Quantitative Flashcards: Exponents Roots

What this deck covers

This deck focuses on Exponents Roots, giving you a quick way to review the definitions, rules, and examples that matter most for GRE Quantitative.

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 value of $\sqrt{49}$?

Answer: $7$. Since $7^2 = 49$ and 7 is nonnegative, it is the principal square root.

Flashcard 2: What is the simplified form of $\sqrt{72}$?

Answer: $6\sqrt{2}$. Factor out perfect square: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$.

Flashcard 3: State the quotient of powers rule for the same base: $\frac{a^m}{a^n}$ equals what (for $a \ne 0$)?

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

Flashcard 4: State the definition of a fractional exponent: $a^{\frac{1}{n}}$ equals what (for $a \ge 0$)?

Answer: $\sqrt[n]{a}$. A fractional exponent with numerator 1 denotes the nth root of the base.

Flashcard 5: What is the simplified form of $\frac{\sqrt{18}}{\sqrt{2}}$ (with $\sqrt{\cdot}$ principal)?

Answer: $3$. Rationalize by combining: $\sqrt{\frac{18}{2}} = \sqrt{9} = 3$.

Flashcard 6: State the power of a product rule: $(ab)^n$ equals what?

Answer: $a^n b^n$. Raising a product to a power distributes the exponent to each factor according to the power of a product rule.

Flashcard 7: What is the simplified form of $\sqrt{\frac{50}{2}}$?

Answer: $5$. Simplify fraction inside: $\frac{50}{2} = 25$, and $\sqrt{25} = 5$.

Flashcard 8: What is the simplified form of $\sqrt{x^2}$ in terms of $x$ for real $x$?

Answer: $|x|$. The principal square root of a square yields the absolute value to ensure nonnegativity.

Flashcard 9: What is the value of $27^{\frac{2}{3}}$?

Answer: $9$. $27 = 3^3$, so $27^{2/3} = (3^3)^{2/3} = 3^2 = 9$ using fractional exponent rules.

Flashcard 10: What is the value of $(2x^3)^2$?

Answer: $4x^6$. Distribute the exponent: $(2)^2 = 4$ and $(x^3)^2 = x^6$.

Flashcard 11: What is the value of $2^3 \cdot 2^5$?

Answer: $2^8$. Add the exponents when multiplying powers with the same base: $3 + 5 = 8$.

Flashcard 12: What is the simplified form of $\frac{a^{-3}b^2}{a^2 b^{-1}}$ (for $a,b \ne 0$)?

Answer: $\frac{b^3}{a^5}$. Combine exponents for like bases: $a^{-3-2} = a^{-5}$ and $b^{2-(-1)} = b^3$, then apply negative exponent rule.

Flashcard 13: State the product of powers rule for the same base: $a^m \cdot a^n$ equals what?

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

Flashcard 14: State the power of a quotient rule: $\left(\frac{a}{b}\right)^n$ equals what (for $b \ne 0$)?

Answer: $\frac{a^n}{b^n}$. Raising a quotient to a power distributes the exponent to both numerator and denominator according to the power of a quotient rule.

Flashcard 15: What is the value of $\sqrt{(-5)^2}$?

Answer: $5$. $(-5)^2 = 25$, and the principal square root of 25 is 5.

Flashcard 16: What is the value of $\sqrt{\frac{1}{81}}$?

Answer: $\frac{1}{9}$. The square root of a fraction is the quotient of the square roots, yielding $\frac{1}{9}$ since $\sqrt{1}=1$ and $\sqrt{81}=9$.

Flashcard 17: State the power of a power rule: $(a^m)^n$ equals what?

Answer: $a^{mn}$. Raising a power to another power multiplies the exponents according to the power of a power rule.

Flashcard 18: State the conversion rule: $a^{\frac{m}{n}}$ equals what (for $a \ge 0$)?

Answer: $\sqrt[n]{a^m}$. A fractional exponent $\frac{m}{n}$ means taking the nth root of the base raised to the m power.

Flashcard 19: What is the value of $(3^2)^4$?

Answer: $3^8$. Multiply the exponents for a power raised to another: $2 \times 4 = 8$.

Flashcard 20: What is the value of $a^0$ for $a \ne 0$?

Answer: $1$. Any non-zero base raised to the zero exponent equals 1 by definition of the zero exponent rule.

Flashcard 21: What is the simplified form of $\frac{x^6}{x^{-2}}$ (for $x \ne 0$)?

Answer: $x^8$. Subtract exponents when dividing: $6 - (-2) = 8$.

Flashcard 22: What is the value of $\frac{5^7}{5^2}$?

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

Flashcard 23: What is the value of $16^{\frac{3}{4}}$?

Answer: $8$. $16 = 2^4$, so $16^{3/4} = (2^4)^{3/4} = 2^3 = 8$ using fractional exponent rules.

Flashcard 24: State the negative exponent rule: $a^{-n}$ equals what (for $a \ne 0$)?

Answer: $\frac{1}{a^n}$. A negative exponent indicates the reciprocal of the base raised to the positive exponent.

Flashcard 25: State the principal square root convention: what does $\sqrt{a}$ mean for $a \ge 0$?

Answer: The nonnegative number whose square is $a$. The principal square root is defined as the nonnegative number that, when squared, gives the original value.