Given x > 0, the expression root(8)(x^(13)) is equivalent to x^(2)root(8)(x^(4)) xroot(8)(x^(5)) xroot(8)(x^(4)) x^(2)root(8)(x^(5))

Understand Expression: Understand the given expression and the properties of exponents and roots. The given expression is root(8)(x^(13)), which means the 8th root of x raised to the 13th power. We can use the property of exponents that states root(n)(a^m) = a^(m/n) to rewrite the expression.Apply Exponent Property: Apply the exponent property to the given expression. Using the property from Step 1, we can rewrite root(8)(x^(13)) as x^(13/8).Simplify Exponent: Simplify the exponent by separating it into integer and fractional parts. The exponent 13/8 can be split into 1 + 5/8, which corresponds to x^(1) * x^(5/8). This is because 13 divided by 8 gives us 1 with a remainder of 5, so 13/8 = 1 + 5/8.Rewrite Using Exponents: Rewrite the expression using the simplified exponents. Now we can express x^(13/8) as x^(1) * x^(5/8), which is the same as x * x^(5/8).Convert Fractional Exponent: Convert the fractional exponent back to root form. The term x^(5/8) can be rewritten as root(8)(x^(5)), using the property from Step 1 in reverse.Combine Terms: Combine the terms to get the final equivalent expression. The final expression is x * root(8)(x^(5)).

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Given
x > 0, the expression
root(8)(x^(13)) is equivalent to

x^(2)root(8)(x^(4))

xroot(8)(x^(5))

xroot(8)(x^(4))

x^(2)root(8)(x^(5))

Given x>0 , the expression $\sqrt[8]{x^{13}}$ is equivalent to $\newline$ $x^{2} \sqrt[8]{x^{4}}$ $\newline$ $x \sqrt[8]{x^{5}}$ $\newline$ $x \sqrt[8]{x^{4}}$ $\newline$ $x^{2} \sqrt[8]{x^{5}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Given

x>0

, the expression

\sqrt[8]{x^{13}}

is equivalent to

\newline

x^{2} \sqrt[8]{x^{4}}

\newline

x \sqrt[8]{x^{5}}

\newline

x \sqrt[8]{x^{4}}

\newline

x^{2} \sqrt[8]{x^{5}}

Understand Expression: Understand the given expression and the properties of exponents and roots. $\newline$ The given expression is $\sqrt[8]{x^{13}}$ , which means the $8$ th root of $x$ raised to the $13$ th power. We can use the property of exponents that states $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ to rewrite the expression.
Apply Exponent Property: Apply the exponent property to the given expression. $\newline$ Using the property from Step $1$ , we can rewrite $\sqrt[8]{x^{13}}$ as $x^{\frac{13}{8}}$ .
Simplify Exponent: Simplify the exponent by separating it into integer and fractional parts. $\newline$ The exponent $\frac{13}{8}$ can be split into $1 + \frac{5}{8}$ , which corresponds to $x^{1} \times x^{\frac{5}{8}}$ . This is because $13$ divided by $8$ gives us $1$ with a remainder of $5$ , so $\frac{13}{8} = 1 + \frac{5}{8}$ .
Rewrite Using Exponents: Rewrite the expression using the simplified exponents. $\newline$ Now we can express $x^{\frac{13}{8}}$ as $x^{1} \times x^{\frac{5}{8}}$ , which is the same as $x \times x^{\frac{5}{8}}$ .
Convert Fractional Exponent: Convert the fractional exponent back to root form. $\newline$ The term $x^{\frac{5}{8}}$ can be rewritten as $\sqrt[8]{x^5}$ , using the property from Step $1$ in reverse.
Combine Terms: Combine the terms to get the final equivalent expression. $\newline$ The final expression is $x \cdot \sqrt[8]{x^{5}}$ .