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Which expression is equivalent to [tex]\frac{(5ab)^3}{30a^{-6}b^{-7}}[/tex]? Assume [tex]a \neq 0, b \neq 0[/tex].

A. [tex]\frac{a^7 b^{10}}{6}[/tex]
B. [tex]\frac{125 a^{18} b^{21}}{30}[/tex]
C. [tex]\frac{25 a^3 b^4}{6}[/tex]
D. [tex]\frac{25 a^9 b^{10}}{6}[/tex]



Answer :

GinnyAnswer
To simplify the given expression [tex]\(\frac{(5ab)^3}{30a^{-6}b^{-7}}\)[/tex], let's go step-by-step:

1. Simplify the Numerator:

[tex]\[ (5ab)^3 = 5^3 \cdot a^3 \cdot b^3 = 125a^3b^3 \][/tex]

So, the new expression is:

[tex]\[ \frac{125a^3b^3}{30a^{-6}b^{-7}} \][/tex]

2. Simplify the Denominator:

[tex]\[ 30a^{-6}b^{-7} \][/tex]

3. Combine the Constants:

[tex]\[ \frac{125}{30} = \frac{25}{6} \quad \text{(since } 125 \div 5 = 25 \text{ and } 30 \div 5 = 6\text{)} \][/tex]

4. Simplify the Exponents for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:

- For [tex]\(a\)[/tex]:
[tex]\[ a^3 \div a^{-6} = a^{3 - (-6)} = a^{3 + 6} = a^9 \][/tex]

- For [tex]\(b\)[/tex]:
[tex]\[ b^3 \div b^{-7} = b^{3 - (-7)} = b^{3 + 7} = b^{10} \][/tex]

5. Combine Everything:

So, the entire simplified expression is:

[tex]\[ \frac{25a^9b^{10}}{6} \][/tex]

Therefore, the expression equivalent to [tex]\(\frac{(5ab)^3}{30a^{-6}b^{-7}}\)[/tex] is:

[tex]\[ \boxed{\frac{25a^9b^{10}}{6}} \][/tex]

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