Given x > 0, the expression root(7)(x^(53)) is equivalent to x^(7) x^(6)root(7)(x^(4)) x^(8) x^(7)root(7)(x^(4))

Understand the expression: Understand the expression root(7)(x^(53)). The expression root(7)(x^(53)) means the 7th root of x raised to the power of 53. We can rewrite this as x^(53/7) using the property that the nth root of a number is the same as raising that number to the power of 1/n.Simplify the exponent: Simplify the exponent. We simplify the fraction 53/7 by performing the division. 53 divided by 7 is 7 with a remainder of 4. So, x^(53/7) can be written as x^(7 + 4/7).Separate the whole number: Separate the whole number and the fraction in the exponent. We can write x^(7 + 4/7) as x^(7) * x^(4/7) using the property that a^(m+n) = a^m * a^n.Recognize the equivalent expression: Recognize the equivalent expression. The expression x^(7) * x^(4/7) can be written as x^(7) * root(7)(x^(4)) because x^(4/7) is the same as the 7th root of x raised to the power of 4.

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Given
x > 0, the expression
root(7)(x^(53)) is equivalent to

x^(7)

x^(6)root(7)(x^(4))

x^(8)

x^(7)root(7)(x^(4))

Given x>0 , the expression $\sqrt[7]{x^{53}}$ is equivalent to $\newline$ $x^{7}$ $\newline$ $x^{6} \sqrt[7]{x^{4}}$ $\newline$ $x^{8}$ $\newline$ $x^{7} \sqrt[7]{x^{4}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Given

x>0

, the expression

\sqrt[7]{x^{53}}

is equivalent to

\newline

x^{7}

\newline

x^{6} \sqrt[7]{x^{4}}

\newline

x^{8}

\newline

x^{7} \sqrt[7]{x^{4}}

Understand the expression: Understand the expression $\sqrt[7]{x^{53}}$ . The expression $\sqrt[7]{x^{53}}$ means the $7$ th root of $x$ raised to the power of $53$ . We can rewrite this as $x^{\frac{53}{7}}$ using the property that the nth root of a number is the same as raising that number to the power of $\frac{1}{n}$ .
Simplify the exponent: Simplify the exponent. $\newline$ We simplify the fraction $\frac{53}{7}$ by performing the division. $\newline$ $53$ divided by $7$ is $7$ with a remainder of $4$ . $\newline$ So, $x^{\frac{53}{7}}$ can be written as $x^{7 + \frac{4}{7}}$ .
Separate the whole number: Separate the whole number and the fraction in the exponent. $\newline$ We can write $x^{7 + \frac{4}{7}}$ as $x^{7} \times x^{\frac{4}{7}}$ using the property that $a^{m+n} = a^m \times a^n$ .
Recognize the equivalent expression: Recognize the equivalent expression. $\newline$ The expression $x^{7} \times x^{\frac{4}{7}}$ can be written as $x^{7} \times \sqrt[7]{x^{4}}$ because $x^{\frac{4}{7}}$ is the same as the $7$ th root of $x$ raised to the power of $4$ .