Let's rewrite the expression [tex]\(24 a^5 b + 6 a b^2\)[/tex] by factoring out the greatest common factor.
We begin by identifying the common factors in each term:
The original expression is:
[tex]\[24 a^5 b + 6 a b^2\][/tex]
Step-by-step solution:
1. Identify Greatest Common Factor (GCF):
- The coefficients are 24 and 6. The GCF of these numbers is 6.
- The variable [tex]\(a\)[/tex] is common in both terms. The highest power of [tex]\(a\)[/tex] that can be factored out is [tex]\(a\)[/tex].
- The variable [tex]\(b\)[/tex] is also common in both terms. The highest power of [tex]\(b\)[/tex] that can be factored out is [tex]\(b\)[/tex].
2. Factor out the GCF:
We factor out [tex]\(6ab\)[/tex] from each term.
Original expression:
[tex]\[24 a^5 b + 6 a b^2\][/tex]
Factoring out [tex]\(6ab\)[/tex]:
[tex]\[6ab \left(\frac{24 a^5 b}{6ab} + \frac{6 a b^2}{6ab}\right)\][/tex]
3. Simplify inside the parentheses:
- For the first term:
[tex]\[\frac{24 a^5 b}{6ab} = 4a^{4}\][/tex]
Explanation:
- [tex]\(24 \div 6 = 4\)[/tex]
- [tex]\(a^5 \div a = a^{4}\)[/tex]
- [tex]\(b \div b = 1\)[/tex]
- For the second term:
[tex]\[\frac{6 a b^2}{6ab} = b\][/tex]
Explanation:
- [tex]\(6 \div 6 = 1\)[/tex]
- [tex]\(a \div a = 1\)[/tex]
- [tex]\(b^2 \div b = b\)[/tex]
Combining these, the simplified expression inside the parentheses is:
[tex]\[4a^4 + b\][/tex]
Thus, the factored form of the original expression is:
[tex]\[6ab \left(4a^4 + b\)\][/tex]
So, the correct rewritten form is:
[tex]\[6ab(4a^4 + b)\][/tex]
This matches option:
[tex]\[6ab(4a^4 + b)\][/tex]
Hence, the correct answer is:
[tex]\[6ab(4a^4 + b)\][/tex]
