Which expression is equivalent to x superscript negative five-thirds

General
1

which expression is equivalent to x superscript negative five-thirds

2

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.

  1. 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}}}

  2. 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}} } }