Sure! Let's simplify each expression step by step.
### Expression 1
[tex]\[
5. \frac{m^2 n^{-5}}{m^7 n^{-17}}
\][/tex]
We can simplify this fraction by using the properties of exponents:
[tex]\[
= 5. \frac{m^2}{m^7} \cdot \frac{n^{-5}}{n^{-17}}
\][/tex]
First, simplify the [tex]\( m \)[/tex] terms:
[tex]\[
\frac{m^2}{m^7} = m^{2-7} = m^{-5}
\][/tex]
Next, simplify the [tex]\( n \)[/tex] terms:
[tex]\[
\frac{n^{-5}}{n^{-17}} = n^{-5 - (-17)} = n^{-5 + 17} = n^{12}
\][/tex]
Putting it all together:
[tex]\[
5. \frac{m^2 n^{-5}}{m^7 n^{-17}} = 5. m^{-5} n^{12} = 5. \frac{n^{12}}{m^5}
\][/tex]
### Expression 2
[tex]\[
\frac{n^{12}}{m^5}
\][/tex]
This expression is already in its simplified form.
### Expression 3
[tex]\[
m^{-3} n^{-10}
\][/tex]
Using the properties of negative exponents, we can rewrite this expression as:
[tex]\[
m^{-3} = \frac{1}{m^3}
\][/tex]
[tex]\[
n^{-10} = \frac{1}{n^{10}}
\][/tex]
So, the expression becomes:
[tex]\[
m^{-3} n^{-10} = \frac{1}{m^3} \cdot \frac{1}{n^{10}} = \frac{1}{m^3 n^{10}}
\][/tex]
### Expression 4
[tex]\[
m^5 n^{12}
\][/tex]
This expression is already in its simplified form.
### Expression 5
[tex]\[
\frac{n^{-3}}{n^{-10}}
\][/tex]
We can simplify the [tex]\( n \)[/tex] terms using the properties of exponents:
[tex]\[
\frac{n^{-3}}{n^{-10}} = n^{-3 - (-10)} = n^{-3 + 10} = n^7
\][/tex]
So, the simplified form is:
[tex]\[
n^7
\][/tex]
### Summary of Simplified Expressions
1. [tex]\( 5. \frac{m^2 n^{-5}}{m^7 n^{-17}} = 5. \frac{n^{12}}{m^5} \)[/tex]
2. [tex]\( \frac{n^{12}}{m^5} \)[/tex]
3. [tex]\( \frac{1}{m^3 n^{10}} \)[/tex]
4. [tex]\( m^5 n^{12} \)[/tex]
5. [tex]\( n^7 \)[/tex]
Each expression has been simplified step by step according to the properties of exponents.
