To solve the equation [tex]\( x^{\frac{1}{5}} = 6 \)[/tex], we need to isolate [tex]\( x \)[/tex].
Here are the detailed steps:
1. Understand the equation: You are given the equation [tex]\( x^{\frac{1}{5}} = 6 \)[/tex]. This means that when [tex]\( x \)[/tex] is raised to the power of [tex]\(\frac{1}{5}\)[/tex], the result is 6.
2. Eliminate the fractional exponent: To get rid of the [tex]\(\frac{1}{5}\)[/tex] exponent on [tex]\( x \)[/tex], you need to raise both sides of the equation to the power of 5. This is because [tex]\( (a^{\frac{1}{5}})^5 = a \)[/tex] for any number [tex]\( a \)[/tex].
3. Apply the power to both sides: Raise both sides of the equation to the power of 5:
[tex]\[
\left(x^{\frac{1}{5}}\right)^5 = 6^5
\][/tex]
4. Simplify the equation: On the left-hand side, raising [tex]\( x^{\frac{1}{5}} \)[/tex] to the power of 5 results in [tex]\( x \)[/tex]:
[tex]\[
x = 6^5
\][/tex]
5. Calculate the right-hand side: Calculate [tex]\( 6^5 \)[/tex]. Here’s the breakdown:
[tex]\[
6^5 = 6 \times 6 \times 6 \times 6 \times 6
\][/tex]
You end up with the following:
[tex]\[
6 \times 6 = 36
\][/tex]
[tex]\[
36 \times 6 = 216
\][/tex]
[tex]\[
216 \times 6 = 1296
\][/tex]
[tex]\[
1296 \times 6 = 7776
\][/tex]
So, the final value of [tex]\( x \)[/tex] is:
[tex]\[
x = 7776
\][/tex]
