Given x > 0, the expression root(6)(x^(43)) is equivalent to x^(7)root(6)(x) x^(8)root(6)(x) x^(8) x^(7)

Understand Expression: Understand the given expression. We need to simplify the sixth root of x raised to the 43rd power, which is written as root(6)(x^(43)).Apply Exponent Property: Apply the property of exponents for roots. The sixth root of x to the power of 43 can be written as x^(43/6).Divide Exponent by 6: Divide the exponent by 6. Dividing 43 by 6 gives us 7 with a remainder of 1, so we can write x^(43/6) as x^(7 + 1/6).Separate Exponent Parts: Separate the integer and fractional parts of the exponent. We can write x^(7 + 1/6) as x^(7) * x^(1/6).Recognize x^(1/6): Recognize that x^(1/6) is the sixth root of x. The expression x^(1/6) is equivalent to the sixth root of x, or root(6)(x).Combine Terms: Combine the terms to get the final expression. The final expression is x^(7) * root(6)(x), which is x^(7)root(6)(x).

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Given
x > 0, the expression
root(6)(x^(43)) is equivalent to

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

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

x^(8)

x^(7)

Given x>0 , the expression $\sqrt[6]{x^{43}}$ is equivalent to $\newline$ $x^{7} \sqrt[6]{x}$ $\newline$ $x^{8} \sqrt[6]{x}$ $\newline$ $x^{8}$ $\newline$ $x^{7}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Given

x>0

, the expression

\sqrt[6]{x^{43}}

is equivalent to

\newline

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

\newline

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

\newline

x^{8}

\newline

x^{7}

Understand Expression: Understand the given expression. $\newline$ We need to simplify the sixth root of $x$ raised to the $43^{\text{rd}}$ power, which is written as $\sqrt[6]{x^{43}}$ .
Apply Exponent Property: Apply the property of exponents for roots. The sixth root of $x$ to the power of $43$ can be written as $x^{(43/6)}$ .
Divide Exponent by $6$ : Divide the exponent by $6$ . $\newline$ Dividing $43$ by $6$ gives us $7$ with a remainder of $1$ , so we can write $x^{(43/6)}$ as $x^{(7 + 1/6)}$ .
Separate Exponent Parts: Separate the integer and fractional parts of the exponent. $\newline$ We can write $x^{7 + \frac{1}{6}}$ as $x^{7} \times x^{\frac{1}{6}}$ .
Recognize $x^{\frac{1}{6}}$ : Recognize that $x^{\frac{1}{6}}$ is the sixth root of $x$ . $\newline$ The expression $x^{\frac{1}{6}}$ is equivalent to the sixth root of $x$ , or $\sqrt[6]{x}$ .
Combine Terms: Combine the terms to get the final expression. $\newline$ The final expression is $x^{7} \cdot \sqrt[6]{x}$ , which is $x^{7}\sqrt[6]{x}$ .