Answer:
Option (b) is correct.
The Equivalent expression to given expression [tex]\sqrt[3]{27x^{18}}[/tex] is [tex]3x^6[/tex]
Step-by-step explanation:
Given : Expression [tex]\sqrt[3]{27x^{18}}[/tex]
We have to choose an equivalent expression to the given expression [tex]\sqrt[3]{27x^{18}}[/tex]
Consider the given expression [tex]\sqrt[3]{27x^{18}}[/tex]
Can be written as [tex]\left(27x^{18}\right)^{\frac{1}{3}}[/tex]
Apply exponent rule [tex]\left(a\cdot \:b\right)^n=a^nb^n[/tex]
We have,
[tex]=27^{\frac{1}{3}}\left(x^{18}\right)^{\frac{1}{3}}[/tex]
Factor 27 as [tex]\:27=3^3[/tex]
[tex]\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}[/tex]
we have,
[tex]\left(3^3\right)^{\frac{1}{3}}=3^{3\cdot \frac{1}{3}}=3[/tex]
Also, For [tex]\left(x^{18}\right)^{\frac{1}{3}}[/tex]
[tex]\mathrm{Apply\:exponent\:rule:\:}\left(a^b\right)^c=a^{bc},\:\quad \mathrm{\:assuming\:}a\ge 0[/tex]
[tex]=x^6[/tex]
Thus, [tex]\left(27x^{18}\right)^{\frac{1}{3}}=3x^6[/tex]
Thus, The Equivalent expression to given expression [tex]\sqrt[3]{27x^{18}}[/tex] is [tex]3x^6[/tex]
