From personal advice to professional guidance, IDNLearn.com has the answers you seek. Our platform offers comprehensive and accurate responses to help you make informed decisions on any topic.
Sagot :
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]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.
