Answer :

We want to simplify the expression

[tex]$$
\left(32 x^{-25} y^{-40}\right)^{-\frac{3}{5}}.
$$[/tex]

Step 1. Rewrite the Expression as a Product Raised to the Exponent

We have a product inside the parentheses. We can apply the power to each factor:

[tex]$$
\left(32 x^{-25} y^{-40}\right)^{-\frac{3}{5}} = 32^{-\frac{3}{5}} \cdot \left(x^{-25}\right)^{-\frac{3}{5}} \cdot \left(y^{-40}\right)^{-\frac{3}{5}}.
$$[/tex]

Step 2. Simplify the Constant Factor

Notice that [tex]$32$[/tex] can be written as [tex]$2^5$[/tex]. Therefore,

[tex]$$
32^{-\frac{3}{5}} = \left(2^5\right)^{-\frac{3}{5}} = 2^{5 \cdot \left(-\frac{3}{5}\right)} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}.
$$[/tex]

Step 3. Simplify the Powers of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]

For the [tex]\( x \)[/tex]-term:

[tex]$$
\left(x^{-25}\right)^{-\frac{3}{5}} = x^{-25 \cdot \left(-\frac{3}{5}\right)} = x^{15}.
$$[/tex]

For the [tex]\( y \)[/tex]-term:

[tex]$$
\left(y^{-40}\right)^{-\frac{3}{5}} = y^{-40 \cdot \left(-\frac{3}{5}\right)} = y^{24}.
$$[/tex]

Step 4. Combine the Results

Now, putting it all together, we have:

[tex]$$
\left(32 x^{-25} y^{-40}\right)^{-\frac{3}{5}} = \frac{1}{8} \cdot x^{15} \cdot y^{24} = \frac{x^{15} y^{24}}{8}.
$$[/tex]

Thus, the expression in its simplest form is:

[tex]$$
\frac{x^{15} y^{24}}{8}.
$$[/tex]

Thanks for taking the time to read Simplify the following expression tex left 32 x 25 y 40 right frac 3 5 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada