Sure, let's solve the equation step-by-step.
Given:
[tex]\[
4^{5x} \div \left(2^{3x}\right)^2 = 256
\][/tex]
1. Simplify the left-hand side of the equation:
Notice that [tex]\(4\)[/tex] can be written as [tex]\(2^2\)[/tex]. Therefore, [tex]\(4^{5x}\)[/tex] can be rewritten in terms of base 2:
[tex]\[
4^{5x} = (2^2)^{5x} = 2^{10x}
\][/tex]
Next, simplify [tex]\(\left(2^{3x}\right)^2\)[/tex]:
[tex]\[
\left(2^{3x}\right)^2 = 2^{3x \cdot 2} = 2^{6x}
\][/tex]
Now the equation becomes:
[tex]\[
\frac{2^{10x}}{2^{6x}} = 256
\][/tex]
2. Apply the properties of exponents:
When you divide powers with the same base, you subtract the exponents:
[tex]\[
\frac{2^{10x}}{2^{6x}} = 2^{10x - 6x} = 2^{4x}
\][/tex]
So the equation now is:
[tex]\[
2^{4x} = 256
\][/tex]
3. Rewrite 256 as a power of 2:
We know that [tex]\(256 = 2^8\)[/tex]:
[tex]\[
2^{4x} = 2^8
\][/tex]
4. Equate the exponents:
Since the bases (2) are the same, we can set the exponents equal to each other:
[tex]\[
4x = 8
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Divide both sides by 4:
[tex]\[
x = \frac{8}{4} = 2
\][/tex]
Therefore, the solution is:
[tex]\[
x = 2
\][/tex]
