Algebra Flashcards: Rewriting Expressions With Radicals Rational Exponents

What this deck covers

This deck focuses on Rewriting Expressions With Radicals Rational Exponents, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

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 $\sqrt[3]{a}\cdot \sqrt[6]{a}$ rewritten as a single rational exponent?

Answer: $a^{\frac{1}{2}}$. Add exponents: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$.

Flashcard 2: What is $27^{\frac{1}{3}}$ rewritten as a radical and simplified?

Answer: $3$. $\sqrt[3]{27} = 3$ since $3^3 = 27$.

Flashcard 3: What is $32^{\frac{2}{5}}$ simplified?

Answer: $4$. $32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^2 = 4$.

Flashcard 4: What is $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{2}{3}}\right)$ simplified?

Answer: $x$. Add exponents: $\frac{1}{3} + \frac{2}{3} = 1$.

Flashcard 5: What is the power of a quotient rule written in exponent form?

Answer: $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$. Distribute the exponent to numerator and denominator.

Flashcard 6: What is the power of a product rule written in exponent form?

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

Flashcard 7: What is $\sqrt[3]{x^2}$ rewritten using a rational exponent?

Answer: $x^{\frac{2}{3}}$. Convert radical to rational exponent form.

Flashcard 8: What is $\frac{\sqrt[3]{x^4}}{\sqrt[3]{x}}$ simplified using rational exponents?

Answer: $x$. Subtract exponents: $\frac{4}{3} - \frac{1}{3} = 1$, so $x^1 = x$.

Flashcard 9: What is $\left(\sqrt[3]{x^2}\right)^{\frac{3}{2}}$ simplified using exponent rules?

Answer: $x$. Apply power rule: $(x^{\frac{2}{3}})^{\frac{3}{2}} = x^1 = x$.

Flashcard 10: What is $64^{\frac{5}{6}}$ simplified?

Answer: $32$. $64^{\frac{5}{6}} = (2^6)^{\frac{5}{6}} = 2^5 = 32$.

Flashcard 11: What is $\left(x^{\frac{2}{5}}\cdot x^{\frac{1}{5}}\right)^5$ simplified?

Answer: $x^3$. Add exponents inside, then apply outer power: $(x^{\frac{3}{5}})^5 = x^3$.

Flashcard 12: What is $\left(x^{-\frac{1}{2}}\right)^4$ simplified?

Answer: $\frac{1}{x^2}$. Multiply exponents: $-\frac{1}{2} \cdot 4 = -2$, so $x^{-2} = \frac{1}{x^2}$

Flashcard 13: What is $(x^{\frac{2}{3}})^3$ simplified?

Answer: $x^2$. Multiply exponents: $\frac{2}{3} \cdot 3 = 2$.

Flashcard 14: What is $\left(\frac{1}{a^{\frac{2}{3}}}\right)\left(a^{\frac{5}{3}}\right)$ simplified?

Answer: $a$. Multiply by reciprocal: add exponents $-\frac{2}{3} + \frac{5}{3} = 1$.

Flashcard 15: What is $\left(\frac{a^{\frac{1}{2}}}{a^{\frac{1}{4}}}\right)$ simplified?

Answer: $a^{\frac{1}{4}}$. Subtract exponents: $\frac{1}{2} - \frac{1}{4} = \frac{1}{4}$.

Flashcard 16: What is $\left(a^{\frac{3}{2}}\right)\left(a^{-\frac{1}{2}}\right)$ simplified?

Answer: $a$. Add exponents: $\frac{3}{2} + (-\frac{1}{2}) = 1$.

Flashcard 17: What is $\sqrt[3]{x^2}\div \sqrt[3]{x^5}$ simplified using exponents?

Answer: $\frac{1}{x}$. Subtract exponents: $\frac{2}{3} - \frac{5}{3} = -1$, so $x^{-1} = \frac{1}{x}$.

Flashcard 18: What is $\frac{x^{\frac{5}{3}}}{x^{\frac{2}{3}}}$ simplified?

Answer: $x$. Subtract exponents: $\frac{5}{3} - \frac{2}{3} = 1$.

Flashcard 19: What is $\frac{1}{x^{\frac{5}{2}}}$ rewritten using a negative exponent?

Answer: $x^{-\frac{5}{2}}$. Reciprocal becomes negative exponent in numerator.

Flashcard 20: What is $\sqrt[4]{x^3}\cdot \sqrt[4]{x}$ simplified using exponents?

Answer: $x$. Add exponents: $\frac{3}{4} + \frac{1}{4} = 1$, so $x^1 = x$.

Flashcard 21: What is $a^{0}$ for $a\ne 0$?

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

Flashcard 22: What is $\sqrt[3]{a^2}\cdot \sqrt[3]{a}$ simplified using exponents?

Answer: $a$. Add exponents: $\frac{2}{3} + \frac{1}{3} = 1$, so $a^1 = a$.

Flashcard 23: What is $\sqrt[8]{x^4}$ rewritten using a simplified rational exponent?

Answer: $x^{\frac{1}{2}}$. Simplify the fraction: $\frac{4}{8} = \frac{1}{2}$.

Flashcard 24: What is $81^{\frac{3}{4}}$ simplified?

Answer: $27$. $81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27$.

Flashcard 25: What is $x^{\frac{7}{3}}\div x^{\frac{1}{3}}$ simplified?

Answer: $x^2$. Subtract exponents: $\frac{7}{3} - \frac{1}{3} = 2$.

Flashcard 26: What is the product rule for exponents written in exponent form for the same base $a$?

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

Flashcard 27: What is $32^{\frac{2}{5}}$ simplified?

Answer: $4$. $32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^2 = 4$.

Flashcard 28: What is $\sqrt[6]{x^3}$ rewritten using a rational exponent and simplified?

Answer: $x^{\frac{1}{2}}$. Simplify the fraction: $\frac{3}{6} = \frac{1}{2}$.

Flashcard 29: What is $16^{\frac{1}{2}}$ rewritten as a radical and simplified?

Answer: $4$. $\sqrt{16} = 4$ since $4^2 = 16$.

Flashcard 30: What is $(x^{\frac{3}{4}})^2$ simplified?

Answer: $x^{\frac{3}{2}}$. Multiply exponents: $\frac{3}{4} \cdot 2 = \frac{3}{2}$.

Flashcard 31: What is $\sqrt[6]{x^3}$ rewritten using a rational exponent and simplified?

Answer: $x^{\frac{1}{2}}$. Simplify the fraction: $\frac{3}{6} = \frac{1}{2}$.

Flashcard 32: What is $\left(a^{\frac{3}{2}}\right)\left(a^{-\frac{1}{2}}\right)$ simplified?

Answer: $a$. Add exponents: $\frac{3}{2} + (-\frac{1}{2}) = 1$.

Flashcard 33: What is $\sqrt{x}\cdot x^{\frac{3}{2}}$ simplified using exponent rules?

Answer: $x^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$.

Flashcard 34: What is $x^{-\frac{2}{3}}$ rewritten without negative exponents?

Answer: $\frac{1}{x^{\frac{2}{3}}}$. Negative exponent moves to denominator as positive.

Flashcard 35: What is $(x^{\frac{2}{3}})^3$ simplified?

Answer: $x^2$. Multiply exponents: $\frac{2}{3} \cdot 3 = 2$.

Flashcard 36: What is $\sqrt[n]{a}$ written using rational exponents (assume $a\ge 0$ and $n\in\mathbb{N}$)?

Answer: $a^{\frac{1}{n}}$. By definition, $\sqrt[n]{a} = a^{\frac{1}{n}}$.

Flashcard 37: What is $\left(\sqrt[3]{x^2}\right)^{\frac{3}{2}}$ simplified using exponent rules?

Answer: $x$. Apply power rule: $(x^{\frac{2}{3}})^{\frac{3}{2}} = x^1 = x$.

Flashcard 38: What is $\sqrt[8]{x^4}$ rewritten using a simplified rational exponent?

Answer: $x^{\frac{1}{2}}$. Simplify the fraction: $\frac{4}{8} = \frac{1}{2}$.

Flashcard 39: What is $\sqrt{x^5}$ rewritten using rational exponents?

Answer: $x^{\frac{5}{2}}$. Convert radical to rational exponent form.

Flashcard 40: What is $\left(\sqrt[5]{a}\right)^3$ rewritten using a rational exponent?

Answer: $a^{\frac{3}{5}}$. Convert radical to rational exponent and apply power.

Flashcard 41: What is $a^{\frac{1}{n}}$ written as a radical (assume $a\ge 0$ and $n\in\mathbb{N}$)?

Answer: $\sqrt[n]{a}$. By definition, $a^{\frac{1}{n}} = \sqrt[n]{a}$.

Flashcard 42: What is $\sqrt[3]{a}\cdot \sqrt[6]{a}$ rewritten as a single rational exponent?

Answer: $a^{\frac{1}{2}}$. Add exponents: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$.

Flashcard 43: What is $\left(\frac{a^{\frac{1}{2}}}{a^{\frac{1}{4}}}\right)$ simplified?

Answer: $a^{\frac{1}{4}}$. Subtract exponents: $\frac{1}{2} - \frac{1}{4} = \frac{1}{4}$.

Flashcard 44: What is $x^{\frac{4}{6}}$ simplified to an equivalent rational exponent in lowest terms?

Answer: $x^{\frac{2}{3}}$. Simplify the fraction: $\frac{4}{6} = \frac{2}{3}$.

Flashcard 45: What is $81^{\frac{3}{4}}$ simplified?

Answer: $27$. $81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27$.

Flashcard 46: What is $\left(\sqrt{x}\right)^3$ rewritten using a rational exponent and simplified?

Answer: $x^{\frac{3}{2}}$. Apply power rule: $(x^{\frac{1}{2}})^3 = x^{\frac{3}{2}}$.

Flashcard 47: What is $\frac{x^{\frac{5}{3}}}{x^{\frac{2}{3}}}$ simplified?

Answer: $x$. Subtract exponents: $\frac{5}{3} - \frac{2}{3} = 1$.

Flashcard 48: What is $\sqrt{a}\cdot \sqrt{a^3}$ simplified using rational exponents?

Answer: $a^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$, so $a^2$.

Flashcard 49: What is $\sqrt[3]{x^2}$ rewritten using a rational exponent?

Answer: $x^{\frac{2}{3}}$. Convert radical to rational exponent form.

Flashcard 50: What is $\left(\sqrt[3]{x}\right)^{-2}$ rewritten without negative exponents?

Answer: $\frac{1}{x^{\frac{2}{3}}}$. Negative exponent moves to denominator as positive.

Flashcard 51: What is $\sqrt[n]{a^m}$ written using rational exponents (assume $a\ge 0$ and $n\in\mathbb{N}$)?

Answer: $a^{\frac{m}{n}}$. By definition, $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.

Flashcard 52: What is $a^{\frac{m}{n}}$ written as a radical (assume $a\ge 0$, $m\in\mathbb{Z}$, $n\in\mathbb{N}$)?

Answer: $\sqrt[n]{a^m}$. By definition, $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Flashcard 53: What is the quotient rule for exponents written in exponent form for the same base $a$?

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

Flashcard 54: What is $\left(x^{-\frac{1}{2}}\right)^4$ simplified?

Answer: $\frac{1}{x^2}$. Multiply exponents: $-\frac{1}{2} \cdot 4 = -2$, so $x^{-2} = \frac{1}{x^2}$

Flashcard 55: What is the power of a power rule for exponents written in exponent form?

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

Flashcard 56: What is $\left(\frac{x^{\frac{1}{2}}}{x^{\frac{3}{2}}}\right)$ simplified?

Answer: $\frac{1}{x}$. Subtract exponents: $\frac{1}{2} - \frac{3}{2} = -1$, so $\frac{1}{x}$.

Flashcard 57: What is the negative exponent rule written in exponent form (assume $a\ne 0$)?

Answer: $a^{-n}=\frac{1}{a^n}$. Negative exponents become reciprocals with positive exponents.

Flashcard 58: What is $\left(x^3\right)^{\frac{1}{2}}$ rewritten using a single rational exponent?

Answer: $x^{\frac{3}{2}}$. Power of a power rule: multiply exponents.

Flashcard 59: What is $a^{0}$ for $a\ne 0$?

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

Flashcard 60: What is $\frac{\sqrt[4]{x}}{\sqrt{x}}$ rewritten using rational exponents and simplified?

Answer: $\frac{1}{x^{\frac{1}{4}}}$. Subtract exponents: $\frac{1}{4} - \frac{1}{2} = -\frac{1}{4}$.

Flashcard 61: What is $x^{\frac{1}{2}}\cdot x^{\frac{3}{2}}$ simplified?

Answer: $x^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$.

Flashcard 62: What is $x^{\frac{7}{3}}\div x^{\frac{1}{3}}$ simplified?

Answer: $x^2$. Subtract exponents: $\frac{7}{3} - \frac{1}{3} = 2$.

Flashcard 63: What is $\left(\frac{1}{a^{\frac{2}{3}}}\right)\left(a^{\frac{5}{3}}\right)$ simplified?

Answer: $a$. Multiply by reciprocal: add exponents $-\frac{2}{3} + \frac{5}{3} = 1$.

Flashcard 64: What is $\left(\sqrt{x}\right)^3$ rewritten using a rational exponent and simplified?

Answer: $x^{\frac{3}{2}}$. Apply power rule: $(x^{\frac{1}{2}})^3 = x^{\frac{3}{2}}$.

Flashcard 65: What is $\sqrt[4]{x^7}$ rewritten using rational exponents?

Answer: $x^{\frac{7}{4}}$. Convert radical to rational exponent form.

Flashcard 66: What is $x^{\frac{4}{6}}$ simplified to an equivalent rational exponent in lowest terms?

Answer: $x^{\frac{2}{3}}$. Simplify the fraction: $\frac{4}{6} = \frac{2}{3}$.

Flashcard 67: What is $\left(\frac{x^{\frac{1}{2}}}{x^{\frac{3}{2}}}\right)$ simplified?

Answer: $\frac{1}{x}$. Subtract exponents: $\frac{1}{2} - \frac{3}{2} = -1$, so $\frac{1}{x}$.

Flashcard 68: What is $a^{\frac{1}{n}}$ written as a radical (assume $a\ge 0$ and $n\in\mathbb{N}$)?

Answer: $\sqrt[n]{a}$. By definition, $a^{\frac{1}{n}} = \sqrt[n]{a}$.

Flashcard 69: What is $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{2}{3}}\right)$ simplified?

Answer: $x$. Add exponents: $\frac{1}{3} + \frac{2}{3} = 1$.

Flashcard 70: What is $\left(\frac{x}{y}\right)^{\frac{3}{2}}$ rewritten as a radical?

Answer: $\sqrt{\left(\frac{x}{y}\right)^3}$. Convert rational exponent to radical form.

Flashcard 71: What is $\sqrt[3]{x^2}\div \sqrt[3]{x^5}$ simplified using exponents?

Answer: $\frac{1}{x}$. Subtract exponents: $\frac{2}{3} - \frac{5}{3} = -1$, so $x^{-1} = \frac{1}{x}$.

Flashcard 72: What is $\left(x^{\frac{2}{5}}\cdot x^{\frac{1}{5}}\right)^5$ simplified?

Answer: $x^3$. Add exponents inside, then apply outer power: $(x^{\frac{3}{5}})^5 = x^3$.