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Answer :
To simplify the expression [tex]\(\sqrt[11]{4x^9 \cdot 9}\)[/tex], let's follow these steps:
1. Combine the Constants:
We can simplify the constants under the root:
[tex]\[
\sqrt[11]{4 \cdot 9 \cdot x^9} = \sqrt[11]{36 \cdot x^9}
\][/tex]
2. Simplifying the Expression:
We can rewrite the expression using the properties of exponents. The eleventh root can be rewritten in exponential form:
[tex]\[
(36 \cdot x^9)^{\frac{1}{11}}
\][/tex]
3. Distribute the Exponents:
Apply the exponent [tex]\(\frac{1}{11}\)[/tex] to both the constant 36 and [tex]\(x^9\)[/tex]:
[tex]\[
36^{\frac{1}{11}} \cdot (x^9)^{\frac{1}{11}}
\][/tex]
4. Simplify the Exponents:
The exponent can be simplified further:
[tex]\[
36^{\frac{1}{11}} \cdot x^{\frac{9}{11}}
\][/tex]
5. Final Expression:
The simplified form of the expression is:
[tex]\[
\sqrt[11]{36} \cdot x^{\frac{9}{11}}
\][/tex]
This is a detailed step-by-step simplification of the expression [tex]\(\sqrt[11]{4x^9 \cdot 9}\)[/tex].
1. Combine the Constants:
We can simplify the constants under the root:
[tex]\[
\sqrt[11]{4 \cdot 9 \cdot x^9} = \sqrt[11]{36 \cdot x^9}
\][/tex]
2. Simplifying the Expression:
We can rewrite the expression using the properties of exponents. The eleventh root can be rewritten in exponential form:
[tex]\[
(36 \cdot x^9)^{\frac{1}{11}}
\][/tex]
3. Distribute the Exponents:
Apply the exponent [tex]\(\frac{1}{11}\)[/tex] to both the constant 36 and [tex]\(x^9\)[/tex]:
[tex]\[
36^{\frac{1}{11}} \cdot (x^9)^{\frac{1}{11}}
\][/tex]
4. Simplify the Exponents:
The exponent can be simplified further:
[tex]\[
36^{\frac{1}{11}} \cdot x^{\frac{9}{11}}
\][/tex]
5. Final Expression:
The simplified form of the expression is:
[tex]\[
\sqrt[11]{36} \cdot x^{\frac{9}{11}}
\][/tex]
This is a detailed step-by-step simplification of the expression [tex]\(\sqrt[11]{4x^9 \cdot 9}\)[/tex].
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