Question

Which expression is equivalent to (256x^16)^1/4? 4x2 4x4 64x2 64x4

Answer 1

$(256x^{16})^\frac{1}{4}=(2^8x^{16})^\frac{1}{4}\\\\Use:\ (ab)^n=a^nb^n\ and\ (a^n)^m=a^{nm}\\\\=(2^8)^\frac{1}{4}(x^{16})^\frac{1}{4}=2^{8\cdot\frac{1}{4}}x^{16\cdot\frac{1}{4}}=2^2x^4=4x^4\\\\Answer:\ \boxed{(256x^{16})^\frac{1}{4}=4x^4}$

Answer 2

Solution:

$=[256x^{16}]^\frac{1}{4}\\\\=[(4 \times 4 \times 4\times 4\times x^{16}]^\frac{1}{4}\\\\ =[(4^4)^{\frac{1}{4}}\times [x^{16}]{\frac{1}{4}}\\\\ =4 \times x^4\\\\={\text{Using the property}} (ab)^\frac{1}{m}=a^\frac{1}{m} b^\frac{1}{m}\\\\{\text{also}} (a^m){\frac{1}{n}}=a^{\frac{m}{n}}$

Factorizing 256,and, $x^{16}$

256= 2×2×2×2×2×2×2×2=4×4×4

$x^{16}=x^4\times x^4\times x^4\times x^4$

Option (B) $4 x^4$ is true.