Answer :
To find the cube root of [tex]\( 8x^{27} \)[/tex], we will break it down into manageable steps:
1. Understanding Cube Roots:
The cube root of a term is found by raising it to the power of [tex]\( \frac{1}{3} \)[/tex].
2. Cube Root of 8:
The number 8 can be expressed as [tex]\( 2^3 \)[/tex]. So, the cube root of 8 is:
[tex]\[
\sqrt[3]{8} = \sqrt[3]{2^3} = 2
\][/tex]
3. Cube Root of [tex]\( x^{27} \)[/tex]:
To find the cube root of a power of [tex]\( x \)[/tex], we divide the exponent by 3. So for [tex]\( x^{27} \)[/tex]:
[tex]\[
\sqrt[3]{x^{27}} = x^{27 \times \frac{1}{3}} = x^9
\][/tex]
4. Combine the Results:
Now, combine the results of the cube root of the numerical coefficient and the cube root of the variable with its power:
[tex]\[
\sqrt[3]{8x^{27}} = 2 \times x^9
\][/tex]
Thus, the cube root of [tex]\( 8x^{27} \)[/tex] is [tex]\( 2x^9 \)[/tex], which matches the answer choice [tex]\( \boxed{2x^9} \)[/tex].
1. Understanding Cube Roots:
The cube root of a term is found by raising it to the power of [tex]\( \frac{1}{3} \)[/tex].
2. Cube Root of 8:
The number 8 can be expressed as [tex]\( 2^3 \)[/tex]. So, the cube root of 8 is:
[tex]\[
\sqrt[3]{8} = \sqrt[3]{2^3} = 2
\][/tex]
3. Cube Root of [tex]\( x^{27} \)[/tex]:
To find the cube root of a power of [tex]\( x \)[/tex], we divide the exponent by 3. So for [tex]\( x^{27} \)[/tex]:
[tex]\[
\sqrt[3]{x^{27}} = x^{27 \times \frac{1}{3}} = x^9
\][/tex]
4. Combine the Results:
Now, combine the results of the cube root of the numerical coefficient and the cube root of the variable with its power:
[tex]\[
\sqrt[3]{8x^{27}} = 2 \times x^9
\][/tex]
Thus, the cube root of [tex]\( 8x^{27} \)[/tex] is [tex]\( 2x^9 \)[/tex], which matches the answer choice [tex]\( \boxed{2x^9} \)[/tex].
