To determine which expression is equivalent to the given expression [tex]\(\frac{20 a^8 b^2}{4 a^2 b}\)[/tex], we need to simplify the expression step-by-step.
### Step 1: Simplify the Coefficients
The fraction has coefficients 20 in the numerator and 4 in the denominator. Dividing these gives:
[tex]\[
\frac{20}{4} = 5
\][/tex]
### Step 2: Simplify the Powers of [tex]\(a\)[/tex]
The numerator has [tex]\(a^8\)[/tex] and the denominator has [tex]\(a^2\)[/tex]. When we divide powers of the same base, we subtract the exponents:
[tex]\[
a^8 \div a^2 = a^{8-2} = a^6
\][/tex]
### Step 3: Simplify the Powers of [tex]\(b\)[/tex]
The numerator has [tex]\(b^2\)[/tex] and the denominator has [tex]\(b\)[/tex] (which is [tex]\(b^1\)[/tex]). Again, we subtract the exponents:
[tex]\[
b^2 \div b^1 = b^{2-1} = b^1 = b
\][/tex]
### Step 4: Combine the Results
Putting it all together, the simplified form of the expression [tex]\(\frac{20 a^8 b^2}{4 a^2 b}\)[/tex] is:
[tex]\[
5 a^6 b
\][/tex]
Thus, the equivalent expression is:
[tex]\[
5 a^6 b
\][/tex]
The correct answer is:
[tex]\[
\boxed{5 a^6 b}
\][/tex]
