Certainly! Let's solve the given expression step-by-step:
[tex]\[
\frac{\left(4^2\right)^2 \times \left(4^3\right)^2}{(64)^2}
\][/tex]
### Step 1: Simplify [tex]\(\left(4^2\right)^2\)[/tex]
First, evaluate [tex]\(4^2\)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
Now, square the result:
[tex]\[
(16)^2 = 256
\][/tex]
So, [tex]\(\left(4^2\right)^2 = 256\)[/tex].
### Step 2: Simplify [tex]\(\left(4^3\right)^2\)[/tex]
Evaluate [tex]\(4^3\)[/tex]:
[tex]\[
4^3 = 64
\][/tex]
Now, square the result:
[tex]\[
(64)^2 = 4096
\][/tex]
So, [tex]\(\left(4^3\right)^2 = 4096\)[/tex].
### Step 3: Simplify [tex]\(64^2\)[/tex]
Evaluate [tex]\(64^2\)[/tex]:
[tex]\[
64^2 = 4096
\][/tex]
### Step 4: Compute the numerator
The numerator of the original expression is:
[tex]\[
\left(4^2\right)^2 \times \left(4^3\right)^2
\][/tex]
Substitute the simplified values:
[tex]\[
256 \times 4096
\][/tex]
Calculate the product:
[tex]\[
256 \times 4096 = 1048576
\][/tex]
So, the numerator is [tex]\(1048576\)[/tex].
### Step 5: Compute the denominator
The denominator of the original expression is:
[tex]\[
(64)^2
\][/tex]
Substitute the simplified value:
[tex]\[
4096
\][/tex]
### Step 6: Compute the final result
Combine the numerator and denominator:
[tex]\[
\frac{1048576}{4096}
\][/tex]
Evaluate the division:
[tex]\[
\frac{1048576}{4096} = 256
\][/tex]
Therefore, the final result of the original expression is:
[tex]\[
256
\][/tex]
So, the step-by-step solution leads to the final result: [tex]\(256\)[/tex].
