Understanding Rational Exponents and Radicals - Algebra
Card 1 of 30
What is the radical-to-exponent translation for $\sqrt[n]{a^m}$ (assume $a>0$)?
What is the radical-to-exponent translation for $\sqrt[n]{a^m}$ (assume $a>0$)?
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$\sqrt[n]{a^m} = a^{\frac{m}{n}}$. The radicand's exponent becomes numerator of rational exponent.
$\sqrt[n]{a^m} = a^{\frac{m}{n}}$. The radicand's exponent becomes numerator of rational exponent.
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What is the power of a product rule for rational exponents (assume $a>0,b>0$)?
What is the power of a product rule for rational exponents (assume $a>0,b>0$)?
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$(ab)^r = a^r b^r$. Distribute the exponent to each factor in the product.
$(ab)^r = a^r b^r$. Distribute the exponent to each factor in the product.
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What is the power-of-a-power rule for rational exponents (assume $a>0$)?
What is the power-of-a-power rule for rational exponents (assume $a>0$)?
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$(a^r)^s = a^{rs}$. When raising a power to a power, multiply exponents.
$(a^r)^s = a^{rs}$. When raising a power to a power, multiply exponents.
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What is the quotient rule for rational exponents (assume $a>0$)?
What is the quotient rule for rational exponents (assume $a>0$)?
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$\frac{a^r}{a^s} = a^{r-s}$. When dividing powers with same base, subtract exponents.
$\frac{a^r}{a^s} = a^{r-s}$. When dividing powers with same base, subtract exponents.
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What is the product rule for rational exponents (assume $a>0$)?
What is the product rule for rational exponents (assume $a>0$)?
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$a^r \cdot a^s = a^{r+s}$. When multiplying powers with same base, add exponents.
$a^r \cdot a^s = a^{r+s}$. When multiplying powers with same base, add exponents.
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Find the simplified form of $\sqrt[6]{a^4}$ using a reduced rational exponent (assume $a>0$).
Find the simplified form of $\sqrt[6]{a^4}$ using a reduced rational exponent (assume $a>0$).
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$a^{\frac{2}{3}}$. $\sqrt[6]{a^4} = a^{\frac{4}{6}} = a^{\frac{2}{3}}$ after reducing the fraction.
$a^{\frac{2}{3}}$. $\sqrt[6]{a^4} = a^{\frac{4}{6}} = a^{\frac{2}{3}}$ after reducing the fraction.
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Find the simplified form of $\left(\sqrt[3]{a}\right)^2$ using rational exponents (assume $a>0$).
Find the simplified form of $\left(\sqrt[3]{a}\right)^2$ using rational exponents (assume $a>0$).
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$a^{\frac{2}{3}}$. $(\sqrt[3]{a})^2 = (a^{\frac{1}{3}})^2 = a^{\frac{2}{3}}$.
$a^{\frac{2}{3}}$. $(\sqrt[3]{a})^2 = (a^{\frac{1}{3}})^2 = a^{\frac{2}{3}}$.
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What is the value of $a^{\frac{m}{n}}$ when $a=0$ and $m>0$ (integer $n\ge 2$)?
What is the value of $a^{\frac{m}{n}}$ when $a=0$ and $m>0$ (integer $n\ge 2$)?
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$0^{\frac{m}{n}} = 0$. Any positive power of zero equals zero.
$0^{\frac{m}{n}} = 0$. Any positive power of zero equals zero.
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Identify the value of $a^{\frac{1}{n}}$ when $a=1$ and integer $n\ge 2$.
Identify the value of $a^{\frac{1}{n}}$ when $a=1$ and integer $n\ge 2$.
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$1^{\frac{1}{n}} = 1$. Any root of 1 equals 1.
$1^{\frac{1}{n}} = 1$. Any root of 1 equals 1.
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What is the value of $(-27)^{\frac{2}{3}}$?
What is the value of $(-27)^{\frac{2}{3}}$?
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$9$. $(-27)^{\frac{2}{3}} = (\sqrt[3]{-27})^2 = (-3)^2 = 9$.
$9$. $(-27)^{\frac{2}{3}} = (\sqrt[3]{-27})^2 = (-3)^2 = 9$.
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What is the value of $(-8)^{\frac{1}{3}}$?
What is the value of $(-8)^{\frac{1}{3}}$?
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$-2$. Cube root of negative number: $\sqrt[3]{-8} = -2$.
$-2$. Cube root of negative number: $\sqrt[3]{-8} = -2$.
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What is the value of $9^{\frac{3}{2}}$?
What is the value of $9^{\frac{3}{2}}$?
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$27$. $9^{\frac{3}{2}} = (\sqrt{9})^3 = 3^3 = 27$.
$27$. $9^{\frac{3}{2}} = (\sqrt{9})^3 = 3^3 = 27$.
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What is the exponent-to-radical translation for $a^{\frac{m}{n}}$ (assume $a>0$)?
What is the exponent-to-radical translation for $a^{\frac{m}{n}}$ (assume $a>0$)?
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$a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Convert rational exponent to radical with index $n$ and radicand $a^m$.
$a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Convert rational exponent to radical with index $n$ and radicand $a^m$.
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What is the radical-to-exponent translation for an $n$th root: $\sqrt[n]{a}$?
What is the radical-to-exponent translation for an $n$th root: $\sqrt[n]{a}$?
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$\sqrt[n]{a} = a^{\frac{1}{n}}$. The $n$th root symbol converts to fractional exponent $\frac{1}{n}$.
$\sqrt[n]{a} = a^{\frac{1}{n}}$. The $n$th root symbol converts to fractional exponent $\frac{1}{n}$.
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What is the power of a quotient rule for rational exponents (assume $a>0,b>0$)?
What is the power of a quotient rule for rational exponents (assume $a>0,b>0$)?
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$\left(\frac{a}{b}\right)^r = \frac{a^r}{b^r}$. Distribute the exponent to both numerator and denominator.
$\left(\frac{a}{b}\right)^r = \frac{a^r}{b^r}$. Distribute the exponent to both numerator and denominator.
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What is the principal-value statement for an even root written as an exponent (assume $a\ge 0$)?
What is the principal-value statement for an even root written as an exponent (assume $a\ge 0$)?
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$a^{\frac{1}{2}}$ is the nonnegative $\sqrt{a}$. Even roots always yield the nonnegative value by definition.
$a^{\frac{1}{2}}$ is the nonnegative $\sqrt{a}$. Even roots always yield the nonnegative value by definition.
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What condition is typically assumed in Algebra $1$ so $a^{\frac{m}{n}}$ is real-valued?
What condition is typically assumed in Algebra $1$ so $a^{\frac{m}{n}}$ is real-valued?
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Assume $a>0$ when $n$ is even. This avoids complex numbers from even roots of negative values.
Assume $a>0$ when $n$ is even. This avoids complex numbers from even roots of negative values.
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What is the simplified form of $(a^{\frac{3}{4}})^2$ (assume $a>0$)?
What is the simplified form of $(a^{\frac{3}{4}})^2$ (assume $a>0$)?
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$a^{\frac{3}{2}}$. Power rule: $(a^{\frac{3}{4}})^2 = a^{\frac{3}{4} \cdot 2} = a^{\frac{6}{4}} = a^{\frac{3}{2}}$.
$a^{\frac{3}{2}}$. Power rule: $(a^{\frac{3}{4}})^2 = a^{\frac{3}{4} \cdot 2} = a^{\frac{6}{4}} = a^{\frac{3}{2}}$.
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What exponent rule motivates defining $a^{\frac{1}{n}}$ as an $n$th root of $a$?
What exponent rule motivates defining $a^{\frac{1}{n}}$ as an $n$th root of $a$?
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Require $(a^{\frac{1}{n}})^n = a$. This ensures consistency with the power rule $(a^r)^s = a^{rs}$.
Require $(a^{\frac{1}{n}})^n = a$. This ensures consistency with the power rule $(a^r)^s = a^{rs}$.
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What is the meaning of $a^{\frac{1}{n}}$ for $a>0$ and integer $n\ge 2$?
What is the meaning of $a^{\frac{1}{n}}$ for $a>0$ and integer $n\ge 2$?
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$a^{\frac{1}{n}} = \sqrt[n]{a}$. The rational exponent $\frac{1}{n}$ denotes the $n$th root.
$a^{\frac{1}{n}} = \sqrt[n]{a}$. The rational exponent $\frac{1}{n}$ denotes the $n$th root.
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What is the meaning of $a^{\frac{m}{n}}$ for $a>0$ and integers $m,n$ with $n\ge 2$?
What is the meaning of $a^{\frac{m}{n}}$ for $a>0$ and integers $m,n$ with $n\ge 2$?
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$a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Take the $n$th root of $a$ raised to the $m$th power.
$a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Take the $n$th root of $a$ raised to the $m$th power.
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What equivalent form of $a^{\frac{m}{n}}$ uses a power of a root (assume $a>0$)?
What equivalent form of $a^{\frac{m}{n}}$ uses a power of a root (assume $a>0$)?
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$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$. Equivalent form: first take the $n$th root, then raise to power $m$.
$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$. Equivalent form: first take the $n$th root, then raise to power $m$.
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What is the value of $a^{\frac{0}{n}}$ for $a\ne 0$ and integer $n\ge 1$?
What is the value of $a^{\frac{0}{n}}$ for $a\ne 0$ and integer $n\ge 1$?
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$a^{\frac{0}{n}} = 1$. Any nonzero number raised to the zero power equals 1.
$a^{\frac{0}{n}} = 1$. Any nonzero number raised to the zero power equals 1.
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What is the meaning of $a^{-\frac{m}{n}}$ for $a>0$ and integers $m,n$ with $n\ge 2$?
What is the meaning of $a^{-\frac{m}{n}}$ for $a>0$ and integers $m,n$ with $n\ge 2$?
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$a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$. Negative exponents create reciprocals of positive exponents.
$a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$. Negative exponents create reciprocals of positive exponents.
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Identify the correct rewrite: $a^{\frac{m}{n}}$ equals which radical form (assume $a>0$)?
Identify the correct rewrite: $a^{\frac{m}{n}}$ equals which radical form (assume $a>0$)?
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$\sqrt[n]{a^m}$. Rational exponent $\frac{m}{n}$ converts to radical $\sqrt[n]{a^m}$.
$\sqrt[n]{a^m}$. Rational exponent $\frac{m}{n}$ converts to radical $\sqrt[n]{a^m}$.
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Identify the correct rewrite: $\sqrt[n]{a^m}$ equals which rational exponent form (assume $a>0$)?
Identify the correct rewrite: $\sqrt[n]{a^m}$ equals which rational exponent form (assume $a>0$)?
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$a^{\frac{m}{n}}$. Radical $\sqrt[n]{a^m}$ converts to rational exponent $\frac{m}{n}$.
$a^{\frac{m}{n}}$. Radical $\sqrt[n]{a^m}$ converts to rational exponent $\frac{m}{n}$.
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What is the simplified form of $\left(\frac{a}{b}\right)^{\frac{1}{3}}$ (assume $a>0,b>0$)?
What is the simplified form of $\left(\frac{a}{b}\right)^{\frac{1}{3}}$ (assume $a>0,b>0$)?
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$\frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}}$. Power of quotient rule distributes the exponent to numerator and denominator.
$\frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}}$. Power of quotient rule distributes the exponent to numerator and denominator.
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What is the simplified form of $\left(a b\right)^{\frac{1}{2}}$ (assume $a>0,b>0$)?
What is the simplified form of $\left(a b\right)^{\frac{1}{2}}$ (assume $a>0,b>0$)?
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$a^{\frac{1}{2}} b^{\frac{1}{2}}$. Power of product rule distributes the exponent to each factor.
$a^{\frac{1}{2}} b^{\frac{1}{2}}$. Power of product rule distributes the exponent to each factor.
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What is the value of $\left(\frac{1}{27}\right)^{-\frac{1}{3}}$?
What is the value of $\left(\frac{1}{27}\right)^{-\frac{1}{3}}$?
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$3$. $(\frac{1}{27})^{-\frac{1}{3}} = 27^{\frac{1}{3}} = 3$.
$3$. $(\frac{1}{27})^{-\frac{1}{3}} = 27^{\frac{1}{3}} = 3$.
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What is the value of $\left(\frac{1}{8}\right)^{\frac{1}{3}}$?
What is the value of $\left(\frac{1}{8}\right)^{\frac{1}{3}}$?
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$\frac{1}{2}$. $(\frac{1}{8})^{\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}} = \frac{1}{2}$.
$\frac{1}{2}$. $(\frac{1}{8})^{\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}} = \frac{1}{2}$.
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