Question

Express the following fraction in simplest form, only using positive exponents.
start fraction, left parenthesis, 3, m, to the power 4 , t, to the power 5 , right parenthesis, squared, divided by, 2, t, to the power 7 , end fraction
2t
7

(3m
4
t
5
)
2

​

Answer 1

To simplify the given expression \(\frac{(3m^4 t^5)^2}{2t^7}\), we will follow these steps:

1. **Square the numerator**: We will expand the numerator \((3m^4 t^5)^2\).
\[
(3m^4 t^5)^2 = 3^2 \cdot (m^4)^2 \cdot (t^5)^2 = 9m^8 t^{10}
\]

2. **Rewrite the fraction**:
\[
\frac{9m^8 t^{10}}{2t^7}
\]

3. **Simplify the fraction**: We will now reduce \(t^{10}\) by \(t^7\):
\[
t^{10} / t^7 = t^{10-7} = t^3
\]

4. **Put the simplified components together**:
\[
\frac{9m^8 t^3}{2}
\]

So, the simplified form of the expression is:
\[
\frac{9m^8 t^3}{2}
\]

This is the final answer, expressed in simplest form using positive exponents.