Answer :
Sure, let's find the Highest Common Factor (HCF) for each of the given expressions step by step.
### (a) [tex]\( 5 a b^2 \)[/tex] and [tex]\( 10 a b^3 \)[/tex]
1. Constants and coefficients:
- The coefficients are [tex]\(5\)[/tex] and [tex]\(10\)[/tex].
- The HCF of [tex]\(5\)[/tex] and [tex]\(10\)[/tex] is [tex]\(5\)[/tex].
2. Variable [tex]\(a\)[/tex]:
- The power of [tex]\(a\)[/tex] in both terms is [tex]\(1\)[/tex].
- So, the HCF for [tex]\(a\)[/tex] is [tex]\(a^1 = a\)[/tex].
3. Variable [tex]\(b\)[/tex]:
- The powers are [tex]\(2\)[/tex] and [tex]\(3\)[/tex].
- The HCF for [tex]\(b\)[/tex] is the lower power, which is [tex]\(b^2\)[/tex].
Combining everything, the HCF is:
[tex]\[ 5a b^2 \][/tex]
### (b) [tex]\( 8 x^2 y \)[/tex], [tex]\( 12 x y \)[/tex], [tex]\( 18 x y^2 \)[/tex]
1. Constants and coefficients:
- The coefficients are [tex]\(8\)[/tex], [tex]\(12\)[/tex], and [tex]\(18\)[/tex].
- The HCF of [tex]\(8\)[/tex], [tex]\(12\)[/tex], and [tex]\(18\)[/tex] is [tex]\(2\)[/tex].
2. Variable [tex]\(x\)[/tex]:
- The powers are [tex]\(2\)[/tex], [tex]\(1\)[/tex], and [tex]\(1\)[/tex].
- The HCF for [tex]\(x\)[/tex] is [tex]\(x^1 = x\)[/tex].
3. Variable [tex]\(y\)[/tex]:
- The powers are [tex]\(1\)[/tex], [tex]\(1\)[/tex], and [tex]\(2\)[/tex].
- The HCF for [tex]\(y\)[/tex] is [tex]\(y^1 = y\)[/tex].
Combining everything, the HCF is:
[tex]\[ 2 x y \][/tex]
### (c) [tex]\(\frac{M g h^2}{k}\)[/tex] and [tex]\(\frac{m G h}{k^2}\)[/tex]
First, let's deal with the terms in the numerator and denominator separately.
1. Numerators:
- In the first term, we have [tex]\(M g h^2\)[/tex].
- In the second term, we have [tex]\(m G h\)[/tex].
For the variables:
- Variable [tex]\(h\)[/tex]: The powers are [tex]\(2\)[/tex] and [tex]\(1\)[/tex]. The HCF is [tex]\(h^1 = h\)[/tex].
2. Denominators:
- The denominators are [tex]\(k\)[/tex] and [tex]\(k^2\)[/tex].
- The HCF for the denominators is [tex]\(k^1 = k\)[/tex].
Combining the HCFs of the numerator and denominator:
[tex]\[ \frac{h}{k} \][/tex]
### (d) [tex]\(\frac{4 x y^2}{9}\)[/tex] and [tex]\(\frac{8 x y}{81}\)[/tex]
1. Numerators:
- The numerators are [tex]\(4 x y^2\)[/tex] and [tex]\(8 x y\)[/tex].
- The coefficients are [tex]\(4\)[/tex] and [tex]\(8\)[/tex]. The HCF of [tex]\(4\)[/tex] and [tex]\(8\)[/tex] is [tex]\(4\)[/tex].
For the variables:
- Variable [tex]\(x\)[/tex]: The powers are [tex]\(1\)[/tex] and [tex]\(1\)[/tex]. The HCF is [tex]\(x^1 = x\)[/tex].
- Variable [tex]\(y\)[/tex]: The powers are [tex]\(2\)[/tex] and [tex]\(1\)[/tex]. The HCF is [tex]\(y^1 = y\)[/tex].
Hence, the HCF of the numerators is:
[tex]\[ 4 x y \][/tex]
2. Denominators:
- The denominators are [tex]\(9\)[/tex] and [tex]\(81\)[/tex].
- The HCF for [tex]\(9\)[/tex] and [tex]\(81\)[/tex] is [tex]\(9\)[/tex].
Combining the HCFs of the numerator and denominator:
[tex]\[ \frac{4 x y}{9} \][/tex]
### Summary:
So, the HCF for each problem is:
(a) [tex]\( 5a b^2 \)[/tex]
(b) [tex]\( 2 x y \)[/tex]
(c) [tex]\(\frac{h}{k} \)[/tex]
(d) [tex]\(\frac{4 x y}{9} \)[/tex]
### (a) [tex]\( 5 a b^2 \)[/tex] and [tex]\( 10 a b^3 \)[/tex]
1. Constants and coefficients:
- The coefficients are [tex]\(5\)[/tex] and [tex]\(10\)[/tex].
- The HCF of [tex]\(5\)[/tex] and [tex]\(10\)[/tex] is [tex]\(5\)[/tex].
2. Variable [tex]\(a\)[/tex]:
- The power of [tex]\(a\)[/tex] in both terms is [tex]\(1\)[/tex].
- So, the HCF for [tex]\(a\)[/tex] is [tex]\(a^1 = a\)[/tex].
3. Variable [tex]\(b\)[/tex]:
- The powers are [tex]\(2\)[/tex] and [tex]\(3\)[/tex].
- The HCF for [tex]\(b\)[/tex] is the lower power, which is [tex]\(b^2\)[/tex].
Combining everything, the HCF is:
[tex]\[ 5a b^2 \][/tex]
### (b) [tex]\( 8 x^2 y \)[/tex], [tex]\( 12 x y \)[/tex], [tex]\( 18 x y^2 \)[/tex]
1. Constants and coefficients:
- The coefficients are [tex]\(8\)[/tex], [tex]\(12\)[/tex], and [tex]\(18\)[/tex].
- The HCF of [tex]\(8\)[/tex], [tex]\(12\)[/tex], and [tex]\(18\)[/tex] is [tex]\(2\)[/tex].
2. Variable [tex]\(x\)[/tex]:
- The powers are [tex]\(2\)[/tex], [tex]\(1\)[/tex], and [tex]\(1\)[/tex].
- The HCF for [tex]\(x\)[/tex] is [tex]\(x^1 = x\)[/tex].
3. Variable [tex]\(y\)[/tex]:
- The powers are [tex]\(1\)[/tex], [tex]\(1\)[/tex], and [tex]\(2\)[/tex].
- The HCF for [tex]\(y\)[/tex] is [tex]\(y^1 = y\)[/tex].
Combining everything, the HCF is:
[tex]\[ 2 x y \][/tex]
### (c) [tex]\(\frac{M g h^2}{k}\)[/tex] and [tex]\(\frac{m G h}{k^2}\)[/tex]
First, let's deal with the terms in the numerator and denominator separately.
1. Numerators:
- In the first term, we have [tex]\(M g h^2\)[/tex].
- In the second term, we have [tex]\(m G h\)[/tex].
For the variables:
- Variable [tex]\(h\)[/tex]: The powers are [tex]\(2\)[/tex] and [tex]\(1\)[/tex]. The HCF is [tex]\(h^1 = h\)[/tex].
2. Denominators:
- The denominators are [tex]\(k\)[/tex] and [tex]\(k^2\)[/tex].
- The HCF for the denominators is [tex]\(k^1 = k\)[/tex].
Combining the HCFs of the numerator and denominator:
[tex]\[ \frac{h}{k} \][/tex]
### (d) [tex]\(\frac{4 x y^2}{9}\)[/tex] and [tex]\(\frac{8 x y}{81}\)[/tex]
1. Numerators:
- The numerators are [tex]\(4 x y^2\)[/tex] and [tex]\(8 x y\)[/tex].
- The coefficients are [tex]\(4\)[/tex] and [tex]\(8\)[/tex]. The HCF of [tex]\(4\)[/tex] and [tex]\(8\)[/tex] is [tex]\(4\)[/tex].
For the variables:
- Variable [tex]\(x\)[/tex]: The powers are [tex]\(1\)[/tex] and [tex]\(1\)[/tex]. The HCF is [tex]\(x^1 = x\)[/tex].
- Variable [tex]\(y\)[/tex]: The powers are [tex]\(2\)[/tex] and [tex]\(1\)[/tex]. The HCF is [tex]\(y^1 = y\)[/tex].
Hence, the HCF of the numerators is:
[tex]\[ 4 x y \][/tex]
2. Denominators:
- The denominators are [tex]\(9\)[/tex] and [tex]\(81\)[/tex].
- The HCF for [tex]\(9\)[/tex] and [tex]\(81\)[/tex] is [tex]\(9\)[/tex].
Combining the HCFs of the numerator and denominator:
[tex]\[ \frac{4 x y}{9} \][/tex]
### Summary:
So, the HCF for each problem is:
(a) [tex]\( 5a b^2 \)[/tex]
(b) [tex]\( 2 x y \)[/tex]
(c) [tex]\(\frac{h}{k} \)[/tex]
(d) [tex]\(\frac{4 x y}{9} \)[/tex]
