Given x > 0, the expression root(9)(x^(77)) is equivalent to x^(9) x^(8)root(9)(x^(6)) x^(8)root(9)(x^(5)) x^(8)

Understand the expression: Understand the expression root(9)(x^(77)). The expression root(9)(x^(77)) means the 9th root of x raised to the 77th power. We can rewrite this as x^(77/9) 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 77/9. To simplify the fraction 77/9, we divide 77 by 9. 77 divided by 9 is 8 with a remainder of 5. So, 77/9 can be written as 8 + 5/9.Rewrite using simplified exponent: Rewrite the expression using the simplified exponent. We can now rewrite x^(77/9) as x^(8 + 5/9). According to the properties of exponents, this is equivalent to x^8 * x^(5/9).Recognize x^(5/9): Recognize that x^(5/9) is the 9th root of x^5. The term x^(5/9) can be rewritten as the 9th root of x^5, which is root(9)(x^5).Combine terms: Combine the terms to get the final expression. The final expression is x^8 * root(9)(x^5), which is one of the given options.

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Given
x > 0, the expression
root(9)(x^(77)) is equivalent to

x^(9)

x^(8)root(9)(x^(6))

x^(8)root(9)(x^(5))

x^(8)

Given x>0 , the expression $\sqrt[9]{x^{77}}$ is equivalent to $\newline$ $x^{9}$ $\newline$ $x^{8} \sqrt[9]{x^{6}}$ $\newline$ $x^{8} \sqrt[9]{x^{5}}$ $\newline$ $x^{8}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Given

x>0

, the expression

\sqrt[9]{x^{77}}

is equivalent to

\newline

x^{9}

\newline

x^{8} \sqrt[9]{x^{6}}

\newline

x^{8} \sqrt[9]{x^{5}}

\newline

x^{8}

Understand the expression: Understand the expression $\sqrt[9]{x^{77}}$ . The expression $\sqrt[9]{x^{77}}$ means the $9$ th root of $x$ raised to the $77$ th power. We can rewrite this as $x^{\frac{77}{9}}$ 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 $\frac{77}{9}$ . To simplify the fraction $\frac{77}{9}$ , we divide $77$ by $9$ . $77$ divided by $9$ is $8$ with a remainder of $5$ . So, $\frac{77}{9}$ can be written as $8 + \frac{5}{9}$ .
Rewrite using simplified exponent: Rewrite the expression using the simplified exponent. $\newline$ We can now rewrite $x^{77/9}$ as $x^{8 + 5/9}$ . $\newline$ According to the properties of exponents, this is equivalent to $x^8 \times x^{5/9}$ .
Recognize $x^{\frac{5}{9}}$ : Recognize that $x^{\frac{5}{9}}$ is the $9$ th root of $x^5$ . The term $x^{\frac{5}{9}}$ can be rewritten as the $9$ th root of $x^5$ , which is $\sqrt[9]{x^5}$ .
Combine terms: Combine the terms to get the final expression. $\newline$ The final expression is $x^8 \cdot \sqrt[9]{x^5}$ , which is one of the given options.