Simplify the expression:

[tex]\[ \frac{m^2 n^{-5}}{m^7 n^{-17}} \][/tex]

A. [tex]\(\frac{n^{12}}{m^5}\)[/tex]

B. [tex]\(m^{-3} n^{-10}\)[/tex]

C. [tex]\(m^5 n^{12}\)[/tex]

D. [tex]\(\frac{n^{-3}}{n^{-10}}\)[/tex]



Answer :

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.