which expression is equivalent to x superscript negative five-thirds
Which expression is equivalent to x^{-\frac{5}{3}} ?
Answer:
To determine an expression equivalent to x^{-\frac{5}{3}} , let’s explore what the negative exponent and fractional exponent signify.
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Negative Exponent: A negative exponent indicates that the base should be taken as the reciprocal. For instance, ( a^{-b} = \frac{1}{a^b} ).
Therefore:
x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} -
Fractional Exponent: A fractional exponent represents a root and a power. Specifically, a^{\frac{m}{n}} = \sqrt[n]{a^m}.
So, x^{\frac{5}{3}} can be rewritten as:
x^{\frac{5}{3}} = \left( x^{5} \right)^{\frac{1}{3}} = \sqrt[3]{x^{5}}Combining these two pieces of information, we have:
x^{-\frac{5}{3}} = \frac{1}{ x^{\frac{5}{3}} } = \frac{1}{ \sqrt[3]{x^{5}} }
Therefore, an equivalent expression for ( x^{-\frac{5}{3}} ) is:
\boxed{ \frac{1}{ \sqrt[3]{x^{5}} } }