Answered

Simplify the expression and write answers using positive exponents only. All variables represent positive real numbers.

[tex]\left(27 a^6 b^{-9}\right)^{2/3}[/tex]

[tex]\left(27 a^6 b^{-9}\right)^{2/3} = \square[/tex]

(Simplify your answer. Use positive exponents only.)



Answer :

GinnyAnswer
To simplify the expression [tex]\(\left(27 a^6 b^{-9}\right)^{2 / 3}\)[/tex] and express the answer with positive exponents only, we can follow these steps:

1. Rewrite the expression inside the parentheses:
[tex]\[ (27 \cdot a^6 \cdot b^{-9})^{2/3} \][/tex]

2. Apply the exponent [tex]\(\frac{2}{3}\)[/tex] to each term inside the parentheses:
[tex]\[ 27^{2/3} \cdot (a^6)^{2/3} \cdot (b^{-9})^{2/3} \][/tex]

3. Simplify each term separately:
- For [tex]\(27^{2/3}\)[/tex]:
[tex]\[ 27^{2/3} = (3^3)^{2/3} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9 \][/tex]

- For [tex]\((a^6)^{2/3}\)[/tex]:
[tex]\[ (a^6)^{2/3} = a^{6 \cdot \frac{2}{3}} = a^4 \][/tex]

- For [tex]\((b^{-9})^{2/3}\)[/tex]:
[tex]\[ (b^{-9})^{2/3} = b^{-9 \cdot \frac{2}{3}} = b^{-6} \][/tex]

4. Combine the simplified terms:
[tex]\[ 9 \cdot a^4 \cdot b^{-6} \][/tex]

5. Convert [tex]\(b^{-6}\)[/tex] to a positive exponent:
[tex]\[ b^{-6} = \frac{1}{b^6} \][/tex]

6. Combine everything into a single fraction:
[tex]\[ 9 \cdot a^4 \cdot \frac{1}{b^6} = \frac{9a^4}{b^6} \][/tex]

Therefore, the simplified expression with positive exponents only is:
[tex]\[ \left(27 a^6 b^{-9}\right)^{2 / 3} = \frac{9a^4}{b^6} \][/tex]

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