Let's go through each division step-by-step:
### (a) Divide [tex]\( 16 m^6 n^5 p^4 \)[/tex] by [tex]\( 8 m^3 n^2 p^2 \)[/tex]
1. Divide the coefficients:
[tex]\[
\frac{16}{8} = 2
\][/tex]
2. Subtract the exponents of like bases:
- For [tex]\( m \)[/tex]:
[tex]\[
6 - 3 = 3
\][/tex]
- For [tex]\( n \)[/tex]:
[tex]\[
5 - 2 = 3
\][/tex]
- For [tex]\( p \)[/tex]:
[tex]\[
4 - 2 = 2
\][/tex]
So the answer is:
[tex]\[
2 m^3 n^3 p^2
\][/tex]
### (b) Divide [tex]\( -24 a b^2 c^3 \)[/tex] by [tex]\( -8 a b c^2 \)[/tex]
1. Divide the coefficients:
[tex]\[
\frac{-24}{-8} = \frac{24}{8} = 3
\][/tex]
2. Subtract the exponents of like bases:
- For [tex]\( a \)[/tex]:
[tex]\[
1 - 1 = 0 \quad (\text{which means } a^0 = 1 \text{ and can be omitted})
\][/tex]
- For [tex]\( b \)[/tex]:
[tex]\[
2 - 1 = 1
\][/tex]
- For [tex]\( c \)[/tex]:
[tex]\[
3 - 2 = 1
\][/tex]
So the answer is:
[tex]\[
3 b c
\][/tex]
### (c) Divide [tex]\( 105 x^8 y^6 \)[/tex] by [tex]\( -15 x^4 y^3 \)[/tex]
1. Divide the coefficients:
[tex]\[
\frac{105}{-15} = -7
\][/tex]
2. Subtract the exponents of like bases:
- For [tex]\( x \)[/tex]:
[tex]\[
8 - 4 = 4
\][/tex]
- For [tex]\( y \)[/tex]:
[tex]\[
6 - 3 = 3
\][/tex]
So the answer is:
[tex]\[
-7 x^4 y^3
\][/tex]
### (d) Divide [tex]\( 36 p^{10} q^9 r^6 \)[/tex] by [tex]\( 6 p^5 q^4 r^3 \)[/tex]
1. Divide the coefficients:
[tex]\[
\frac{36}{6} = 6
\][/tex]
2. Subtract the exponents of like bases:
- For [tex]\( p \)[/tex]:
[tex]\[
10 - 5 = 5
\][/tex]
- For [tex]\( q \)[/tex]:
[tex]\[
9 - 4 = 5
\][/tex]
- For [tex]\( r \)[/tex]:
[tex]\[
6 - 3 = 3
\][/tex]
So the answer is:
[tex]\[
6 p^5 q^5 r^3
\][/tex]
In summary, the results are:
- (a) [tex]\( 2 m^3 n^3 p^2 \)[/tex]
- (b) [tex]\( 3 b c \)[/tex]
- (c) [tex]\( -7 x^4 y^3 \)[/tex]
- (d) [tex]\( 6 p^5 q^5 r^3 \)[/tex]
