Answer :

We start with the function

[tex]$$
f(x)=4\sqrt{8x^{10}+10x^9}.
$$[/tex]

Step 1. Factor the radicand

Notice that both terms inside the square root have a common factor of [tex]$x^9$[/tex]. Factor this out:

[tex]$$
8x^{10}+10x^9 = x^9(8x+10).
$$[/tex]

The expression [tex]$8x+10$[/tex] can also be factored by taking out a factor of [tex]$2$[/tex]:

[tex]$$
8x+10 = 2(4x+5).
$$[/tex]

Thus, the radicand becomes:

[tex]$$
8x^{10}+10x^9 = 2x^9(4x+5).
$$[/tex]

Step 2. Separate the square roots

Now rewrite [tex]$f(x)$[/tex] using the factored form:

[tex]$$
f(x)=4\sqrt{2x^9(4x+5)}.
$$[/tex]

By the property of square roots, [tex]$\sqrt{ab}=\sqrt{a}\sqrt{b}$[/tex], we can express this as:

[tex]$$
f(x)=4\sqrt{2}\sqrt{x^9}\sqrt{4x+5}.
$$[/tex]

Step 3. Simplify [tex]$\sqrt{x^9}$[/tex]

Since [tex]$x^9$[/tex] can be written as [tex]$x^9 = x^8 \cdot x$[/tex] and [tex]$x^8$[/tex] is a perfect square (because [tex]$x^8=(x^4)^2$[/tex]), we have:

[tex]$$
\sqrt{x^9} = \sqrt{x^8}\sqrt{x} = x^4\sqrt{x}.
$$[/tex]

Step 4. Write the final simplified expression

Substitute the simplified square root back into the expression for [tex]$f(x)$[/tex]:

[tex]$$
f(x)=4\sqrt{2}\, \Bigl( x^4\sqrt{x} \Bigr)\sqrt{4x+5}.
$$[/tex]

This simplifies to:

[tex]$$
f(x)=4\sqrt{2}\, x^4 \sqrt{x(4x+5)}.
$$[/tex]

An equivalent form is obtained by combining the exponents on [tex]$x$[/tex]:

Since [tex]$\sqrt{x^9} = x^{9/2}$[/tex], we can also write

[tex]$$
f(x)=4\sqrt{2}\, x^{9/2}\sqrt{4x+5}.
$$[/tex]

Thus, the final simplified form of the function is

[tex]$$
\boxed{4\sqrt{2}\, x^{9/2}\sqrt{4x+5}}.
$$[/tex]

Thanks for taking the time to read Given the function tex f x 4 sqrt 8x 10 10x 9 tex identify any terms with negative exponents. 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