Rewriting Expressions with Radicals, Rational Exponents - Algebra
Card 1 of 30
What is $\sqrt{a}\cdot \sqrt{a^3}$ simplified using rational exponents?
What is $\sqrt{a}\cdot \sqrt{a^3}$ simplified using rational exponents?
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$a^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$, so $a^2$.
$a^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$, so $a^2$.
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What is $x^{\frac{4}{6}}$ simplified to an equivalent rational exponent in lowest terms?
What is $x^{\frac{4}{6}}$ simplified to an equivalent rational exponent in lowest terms?
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$x^{\frac{2}{3}}$. Simplify the fraction: $\frac{4}{6} = \frac{2}{3}$.
$x^{\frac{2}{3}}$. Simplify the fraction: $\frac{4}{6} = \frac{2}{3}$.
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What is $a^{\frac{1}{n}}$ written as a radical (assume $a\ge 0$ and $n\in\mathbb{N}$)?
What is $a^{\frac{1}{n}}$ written as a radical (assume $a\ge 0$ and $n\in\mathbb{N}$)?
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$\sqrt[n]{a}$. By definition, $a^{\frac{1}{n}} = \sqrt[n]{a}$.
$\sqrt[n]{a}$. By definition, $a^{\frac{1}{n}} = \sqrt[n]{a}$.
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What is $a^{\frac{m}{n}}$ written as a radical (assume $a\ge 0$, $m\in\mathbb{Z}$, $n\in\mathbb{N}$)?
What is $a^{\frac{m}{n}}$ written as a radical (assume $a\ge 0$, $m\in\mathbb{Z}$, $n\in\mathbb{N}$)?
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$\sqrt[n]{a^m}$. By definition, $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.
$\sqrt[n]{a^m}$. By definition, $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.
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What is $\sqrt[n]{a}$ written using rational exponents (assume $a\ge 0$ and $n\in\mathbb{N}$)?
What is $\sqrt[n]{a}$ written using rational exponents (assume $a\ge 0$ and $n\in\mathbb{N}$)?
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$a^{\frac{1}{n}}$. By definition, $\sqrt[n]{a} = a^{\frac{1}{n}}$.
$a^{\frac{1}{n}}$. By definition, $\sqrt[n]{a} = a^{\frac{1}{n}}$.
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What is $\sqrt[n]{a^m}$ written using rational exponents (assume $a\ge 0$ and $n\in\mathbb{N}$)?
What is $\sqrt[n]{a^m}$ written using rational exponents (assume $a\ge 0$ and $n\in\mathbb{N}$)?
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$a^{\frac{m}{n}}$. By definition, $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.
$a^{\frac{m}{n}}$. By definition, $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.
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What is the product rule for exponents written in exponent form for the same base $a$?
What is the product rule for exponents written in exponent form for the same base $a$?
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$a^m\cdot a^n=a^{m+n}$. When multiplying same bases, add the exponents.
$a^m\cdot a^n=a^{m+n}$. When multiplying same bases, add the exponents.
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What is the quotient rule for exponents written in exponent form for the same base $a$?
What is the quotient rule for exponents written in exponent form for the same base $a$?
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$\frac{a^m}{a^n}=a^{m-n}$. When dividing same bases, subtract the exponents.
$\frac{a^m}{a^n}=a^{m-n}$. When dividing same bases, subtract the exponents.
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What is the power of a power rule for exponents written in exponent form?
What is the power of a power rule for exponents written in exponent form?
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$(a^m)^n=a^{mn}$. When raising a power to a power, multiply exponents.
$(a^m)^n=a^{mn}$. When raising a power to a power, multiply exponents.
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What is the power of a product rule written in exponent form?
What is the power of a product rule written in exponent form?
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$(ab)^n=a^n b^n$. Distribute the exponent to each factor in the product.
$(ab)^n=a^n b^n$. Distribute the exponent to each factor in the product.
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What is the power of a quotient rule written in exponent form?
What is the power of a quotient rule written in exponent form?
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$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$. Distribute the exponent to numerator and denominator.
$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$. Distribute the exponent to numerator and denominator.
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What is the negative exponent rule written in exponent form (assume $a\ne 0$)?
What is the negative exponent rule written in exponent form (assume $a\ne 0$)?
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$a^{-n}=\frac{1}{a^n}$. Negative exponents become reciprocals with positive exponents.
$a^{-n}=\frac{1}{a^n}$. Negative exponents become reciprocals with positive exponents.
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What is $a^{0}$ for $a\ne 0$?
What is $a^{0}$ for $a\ne 0$?
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$1$. Any nonzero base to the zero power equals 1.
$1$. Any nonzero base to the zero power equals 1.
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What is $16^{\frac{1}{2}}$ rewritten as a radical and simplified?
What is $16^{\frac{1}{2}}$ rewritten as a radical and simplified?
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$4$. $\sqrt{16} = 4$ since $4^2 = 16$.
$4$. $\sqrt{16} = 4$ since $4^2 = 16$.
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What is $27^{\frac{1}{3}}$ rewritten as a radical and simplified?
What is $27^{\frac{1}{3}}$ rewritten as a radical and simplified?
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$3$. $\sqrt[3]{27} = 3$ since $3^3 = 27$.
$3$. $\sqrt[3]{27} = 3$ since $3^3 = 27$.
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What is $81^{\frac{3}{4}}$ simplified?
What is $81^{\frac{3}{4}}$ simplified?
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$27$. $81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27$.
$27$. $81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27$.
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What is $32^{\frac{2}{5}}$ simplified?
What is $32^{\frac{2}{5}}$ simplified?
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$4$. $32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^2 = 4$.
$4$. $32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^2 = 4$.
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What is $64^{\frac{5}{6}}$ simplified?
What is $64^{\frac{5}{6}}$ simplified?
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$32$. $64^{\frac{5}{6}} = (2^6)^{\frac{5}{6}} = 2^5 = 32$.
$32$. $64^{\frac{5}{6}} = (2^6)^{\frac{5}{6}} = 2^5 = 32$.
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What is $x^{\frac{1}{2}}\cdot x^{\frac{3}{2}}$ simplified?
What is $x^{\frac{1}{2}}\cdot x^{\frac{3}{2}}$ simplified?
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$x^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$.
$x^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$.
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What is $x^{\frac{7}{3}}\div x^{\frac{1}{3}}$ simplified?
What is $x^{\frac{7}{3}}\div x^{\frac{1}{3}}$ simplified?
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$x^2$. Subtract exponents: $\frac{7}{3} - \frac{1}{3} = 2$.
$x^2$. Subtract exponents: $\frac{7}{3} - \frac{1}{3} = 2$.
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What is $(x^{\frac{2}{3}})^3$ simplified?
What is $(x^{\frac{2}{3}})^3$ simplified?
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$x^2$. Multiply exponents: $\frac{2}{3} \cdot 3 = 2$.
$x^2$. Multiply exponents: $\frac{2}{3} \cdot 3 = 2$.
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What is $(x^{\frac{3}{4}})^2$ simplified?
What is $(x^{\frac{3}{4}})^2$ simplified?
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$x^{\frac{3}{2}}$. Multiply exponents: $\frac{3}{4} \cdot 2 = \frac{3}{2}$.
$x^{\frac{3}{2}}$. Multiply exponents: $\frac{3}{4} \cdot 2 = \frac{3}{2}$.
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What is $\left(x^3\right)^{\frac{1}{2}}$ rewritten using a single rational exponent?
What is $\left(x^3\right)^{\frac{1}{2}}$ rewritten using a single rational exponent?
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$x^{\frac{3}{2}}$. Power of a power rule: multiply exponents.
$x^{\frac{3}{2}}$. Power of a power rule: multiply exponents.
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What is $\sqrt[3]{x^2}$ rewritten using a rational exponent?
What is $\sqrt[3]{x^2}$ rewritten using a rational exponent?
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$x^{\frac{2}{3}}$. Convert radical to rational exponent form.
$x^{\frac{2}{3}}$. Convert radical to rational exponent form.
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What is $\sqrt{x^5}$ rewritten using rational exponents?
What is $\sqrt{x^5}$ rewritten using rational exponents?
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$x^{\frac{5}{2}}$. Convert radical to rational exponent form.
$x^{\frac{5}{2}}$. Convert radical to rational exponent form.
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What is $\sqrt[4]{x^7}$ rewritten using rational exponents?
What is $\sqrt[4]{x^7}$ rewritten using rational exponents?
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$x^{\frac{7}{4}}$. Convert radical to rational exponent form.
$x^{\frac{7}{4}}$. Convert radical to rational exponent form.
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What is $x^{-\frac{2}{3}}$ rewritten without negative exponents?
What is $x^{-\frac{2}{3}}$ rewritten without negative exponents?
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$\frac{1}{x^{\frac{2}{3}}}$. Negative exponent moves to denominator as positive.
$\frac{1}{x^{\frac{2}{3}}}$. Negative exponent moves to denominator as positive.
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What is $\frac{1}{x^{\frac{5}{2}}}$ rewritten using a negative exponent?
What is $\frac{1}{x^{\frac{5}{2}}}$ rewritten using a negative exponent?
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$x^{-\frac{5}{2}}$. Reciprocal becomes negative exponent in numerator.
$x^{-\frac{5}{2}}$. Reciprocal becomes negative exponent in numerator.
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What is $\left(\frac{x}{y}\right)^{\frac{3}{2}}$ rewritten as a radical?
What is $\left(\frac{x}{y}\right)^{\frac{3}{2}}$ rewritten as a radical?
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$\sqrt{\left(\frac{x}{y}\right)^3}$. Convert rational exponent to radical form.
$\sqrt{\left(\frac{x}{y}\right)^3}$. Convert rational exponent to radical form.
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What is $\left(\sqrt[5]{a}\right)^3$ rewritten using a rational exponent?
What is $\left(\sqrt[5]{a}\right)^3$ rewritten using a rational exponent?
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$a^{\frac{3}{5}}$. Convert radical to rational exponent and apply power.
$a^{\frac{3}{5}}$. Convert radical to rational exponent and apply power.
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