Divide the following monomials by the given monomials:

(a) [tex]\(\frac{16 m^6 n^5 p^4}{8 m^3 n^2 p^2}\)[/tex]

(b) [tex]\(\frac{-24 a b^2 c^3}{-8 a b c^2}\)[/tex]

(c) [tex]\(\frac{105 x^8 y^6}{-15 x^4 y^3}\)[/tex]

(d) [tex]\(\frac{36 p^{10} q^9 r^6}{6 p^5 q^4 r^3}\)[/tex]



Answer :

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]