Given expression: [tex](256x^{16})^{1/4}[/tex].
[tex]\mathrm{Apply\:exponent\:rule}:\quad \left(a\cdot \:b\right)^n=a^nb^n[/tex]
[tex]=256^{\frac{1}{4}}\left(x^{16}\right)^{\frac{1}{4}}[/tex]
[tex]256=4^4[/tex]
[tex]256^{\frac{1}{4}}=\left(4^4\right)^{\frac{1}{4}}=4[/tex]
[tex]\left(x^{16}\right)^{\frac{1}{4}}=x^{16\cdot \frac{1}{4}}=x^4[/tex]
[tex](256x^{16})^{1/4} =4x^4[/tex]
Answer: The correct option is (B) [tex]4x^4.[/tex]
Step-by-step explanation: We are given to select the correct expression that is equivalent to the expression below:
[tex]E=(256x^{16})^\frac{1}{4}.[/tex]
We will be using the following properties of exponents:
[tex](i)~(a^b)^c=a^{b\times c},\\\\(ii)~(ab)^c=a^cb^c.[/tex]
We have
[tex]E\\\\=(256x^{16})^\frac{1}{4}\\\\=(4^4x^{16})^\frac{1}{4}\\\\=(4^4)^\frac{1}{4}(x^{16}^\frac{1}{4})\\\\=4^{4\times\frac{1}{4}}x^{16\times\frac{1}{4}}\\\\=4x^4.[/tex]
Therefore, the required equivalent expression is [tex]4x^4.[/tex]
Thus, (B) is the correct option.
