Given x > 0, the expression root(7)(x^(10)) is equivalent to x^(2)root(7)(x^(3)) xroot(7)(x^(3)) x^(2) x

Given expression: We are given the expression root(7)(x^(10)) and we need to simplify it. The 7th root of x to the 10th power can be written as x^(10/7).Splitting the exponent: Now, we can split the exponent 10/7 into two parts: one that is a whole number and one that is a fraction less than 1. 10/7 can be split into 1 + 3/7 because 1 is the whole number part of 10/7 and 3/7 is the fractional part. So, x^(10/7) = x^(1 + 3/7).Using exponent properties: Using the property of exponents that says a^(m+n) = a^m * a^n, we can write x^(1 + 3/7) as x^1 * x^(3/7). This means x^(10/7) = x * x^(3/7).Simplifying x^(3/7): Now, x^(3/7) is the 7th root of x cubed, which can be written as root(7)(x^3). So, the expression x * x^(3/7) can be written as x * root(7)(x^3).Final equivalent expression: Therefore, the equivalent expression for root(7)(x^(10)) given x > 0 is x * root(7)(x^3). This matches one of the options provided.

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Given
x > 0, the expression
root(7)(x^(10)) is equivalent to

x^(2)root(7)(x^(3))

xroot(7)(x^(3))

x^(2)

x

Given x>0 , the expression $\sqrt[7]{x^{10}}$ is equivalent to $\newline$ $x^{2} \sqrt[7]{x^{3}}$ $\newline$ $x \sqrt[7]{x^{3}}$ $\newline$ $x^{2}$ $\newline$ $x$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Given

x>0

, the expression

\sqrt[7]{x^{10}}

is equivalent to

\newline

x^{2} \sqrt[7]{x^{3}}

\newline

x \sqrt[7]{x^{3}}

\newline

x^{2}

\newline

x

Given expression: We are given the expression $\sqrt[7]{x^{10}}$ and we need to simplify it. $\newline$ The $7$ th root of $x$ to the $10$ th power can be written as $x^{\frac{10}{7}}$ .
Splitting the exponent: Now, we can split the exponent $\frac{10}{7}$ into two parts: one that is a whole number and one that is a fraction less than $1$ . $\newline$ $\frac{10}{7}$ can be split into $1 + \frac{3}{7}$ because $1$ is the whole number part of $\frac{10}{7}$ and $\frac{3}{7}$ is the fractional part. $\newline$ So, $x^{\frac{10}{7}} = x^{1 + \frac{3}{7}}$ .
Using exponent properties: Using the property of exponents that says $a^{m+n} = a^m \cdot a^n$ , we can write $x^{1 + \frac{3}{7}}$ as $x^1 \cdot x^{\frac{3}{7}}$ . This means $x^{\frac{10}{7}} = x \cdot x^{\frac{3}{7}}$ .
Simplifying $x^{\frac{3}{7}}$ : Now, $x^{\frac{3}{7}}$ is the $7$ th root of $x$ cubed, which can be written as $\sqrt[7]{x^3}$ . So, the expression $x \cdot x^{\frac{3}{7}}$ can be written as $x \cdot \sqrt[7]{x^3}$ .
Final equivalent expression: Therefore, the equivalent expression for $\sqrt[7]{x^{10}}$ given x > 0 is $x \cdot \sqrt[7]{x^3}$ . This matches one of the options provided.