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
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
Answers
GPT-4o mini
To simplify the fraction \(\frac{(3m^4 t^5)^2}{2t^7}\), we'll start by squaring the expression in the numerator and then simplifying.
1. **Square the numerator**:
\[
(3m^4 t^5)^2 = 3^2 (m^4)^2 (t^5)^2 = 9m^8 t^{10}
\]
So now our fraction looks like:
\[
\frac{9m^8 t^{10}}{2t^7}
\]
2. **Simplify the fraction**:
When simplifying, we can divide \(t^{10}\) by \(t^7\):
\[
t^{10} / t^7 = t^{10-7} = t^3
\]
Now our fraction becomes:
\[
\frac{9m^8 t^3}{2}
\]
3. **Express the result**:
We can express the simplified fraction neatly as:
\[
\frac{9m^8 t^3}{2}
\]
Thus, the final answer in simplest form using only positive exponents is:
\[
\frac{9m^8 t^3}{2}
\]
1. **Square the numerator**:
\[
(3m^4 t^5)^2 = 3^2 (m^4)^2 (t^5)^2 = 9m^8 t^{10}
\]
So now our fraction looks like:
\[
\frac{9m^8 t^{10}}{2t^7}
\]
2. **Simplify the fraction**:
When simplifying, we can divide \(t^{10}\) by \(t^7\):
\[
t^{10} / t^7 = t^{10-7} = t^3
\]
Now our fraction becomes:
\[
\frac{9m^8 t^3}{2}
\]
3. **Express the result**:
We can express the simplified fraction neatly as:
\[
\frac{9m^8 t^3}{2}
\]
Thus, the final answer in simplest form using only positive exponents is:
\[
\frac{9m^8 t^3}{2}
\]