Question

Express the following fraction in simplest form using only positive exponents.
start fraction, 20, b, divided by, left parenthesis, 4, b, cubed, right parenthesis, cubed, end fraction
(4b
3
)
3

20b
​

Answer 1

To simplify the expression \(\frac{20b}{(4b^3)^3}\), we will first simplify the denominator:

1. Start by simplifying the denominator \((4b^3)^3\):
\[
(4b^3)^3 = 4^3 \cdot (b^3)^3 = 64b^9
\]

2. Now substitute this back into the fraction:
\[
\frac{20b}{(4b^3)^3} = \frac{20b}{64b^9}
\]

3. Next, we can simplify the fraction by separating the coefficients and the variables:
\[
\frac{20}{64} \cdot \frac{b}{b^9} = \frac{20}{64} \cdot b^{1-9} = \frac{20}{64} \cdot b^{-8}
\]

4. The coefficient \(\frac{20}{64}\) can be simplified:
\[
\frac{20}{64} = \frac{5}{16} \quad (\text{divide numerator and denominator by 4})
\]

5. Therefore, we have:
\[
\frac{5}{16} \cdot b^{-8} = \frac{5}{16b^8}
\]

This means our final expression in simplest form using only positive exponents is:
\[
\frac{5}{16b^8}
\]