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]
[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]
