Question

Which expression is equivalent to
4^sqrt 6 / 3^sqrt2

Answer 1

Answer:

$\dfrac{\sqrt[12]{55296} }{2}$

Step-by-step explanation:

Given expression:

$\dfrac{\sqrt[4]{6}}{\sqrt[3]{2}}$

$\textsf{Apply exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:$

$\implies \dfrac{\sqrt[4]{6}}{\sqrt[3]{2}}=\dfrac{6^{\frac{1}{4}}}{2^{\frac{1}{3}}}$

Multiply the numerator and denominator by $2^{\frac{2}{3}}$ :

$\implies \dfrac{6^{\frac{1}{4}}}{2^{\frac{1}{3}}} \times \dfrac{2^{\frac{2}{3}}}{2^{\frac{2}{3}}}$

$\textsf{Apply exponent rule to the denominator} \quad a^b \cdot a^c=a^{b+c}:$

$\implies \dfrac{6^{\frac{1}{4}}}{2^{\frac{1}{3}}} \times \dfrac{2^{\frac{2}{3}}}{2^{\frac{2}{3}}}=\dfrac{6^{\frac{1}{4}} \cdot 2^{\frac{2}{3}}}{2^{\frac{1}{3}+\frac{2}{3}}}=\dfrac{6^{\frac{1}{4}} \cdot 2^{\frac{2}{3}}}{2}$

Rewrite 1/4 as 3/12 and 2/3 as 8/12 :

$\implies \dfrac{6^{\frac{1}{4}} \cdot 2^{\frac{2}{3}}}{2}=\dfrac{6^{\frac{3}{12}} \cdot 2^{\frac{8}{12}}}{2}$

$\textsf{Apply exponent rule} \quad a^c \cdot b^c=(a \cdot b)^c:$

$\implies \dfrac{6^{\frac{3}{12}} \cdot 2^{\frac{8}{12}}}{2}=\dfrac{(6^3 \cdot 2^{8})^\frac{1}{12}}{2}$

Simplify the operation in the parentheses:

$\implies \dfrac{(6^3 \cdot 2^{8})^\frac{1}{12}}{2}=\dfrac{(216\cdot 256)^\frac{1}{12}}{2}=\dfrac{(55296)^\frac{1}{12}}{2}$

$\textsf{Finally, apply exponent rule} \quad a^{\frac{1}{n}}=\sqrt[n]{a}:$

$\implies \dfrac{(55296)^\frac{1}{12}}{2}=\dfrac{\sqrt[12]{55296} }{2}$

Answer 2

$\\ \rm\Rrightarrow \dfrac{\sqrt[4]{6}}{\sqrt[3]{2}}$

• ^b√a=a^1/b

$\\ \rm\Rrightarrow \dfrac{6^{\dfrac{1}{4}}}{2^{\dfrac{1}{3}}}$

• 6=2×3

$\\ \rm\Rrightarrow \dfrac{2^{\dfrac{1}{4}}3^{\dfrac{1}{4}}}{2^{\dfrac{1}{3}}}$

• a^m÷a^n=a^m-n

$\\ \rm\Rrightarrow 2^{\dfrac{1}{4}-\dfrac{1}{3}}3^{\dfrac{1}{4}}$

$\\ \rm\Rrightarrow 2^{\dfrac{-1}{12}}3^{\dfrac{1}{4}}$

$\\ \rm\Rrightarrow \dfrac{3^{\dfrac{1}{4}}}{2^{\dfrac{1}{12}}}$

• Equalise exponential denominators

$\\ \rm\Rrightarrow \dfrac{3^{\dfrac{3}{12}}}{2^{\dfrac{1}{12}}}$

$\\ \rm\Rrightarrow \left(\dfrac{3^3}{2}\right)^{\dfrac{1}{12}}$

$\\ \rm\Rrightarrow \sqrt[12]{\dfrac{3^3}{2}}$

$\\ \rm\Rrightarrow \sqrt[12]{\dfrac{3^3\times 2^82^3}{22^82^3}}$

$\\ \rm\Rrightarrow \sqrt[12]{\dfrac{27(256)(8)}{2^12}}$

$\\ \rm\Rrightarrow \dfrac{\sqrt[12]{6912(8)}}{2}$

$\\ \rm\Rrightarrow \dfrac{\sqrt[12]{55296}}{2}$