Question

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

Answer 1

`\\ \rm\Rrightarrow \frac{\sqrt[4]{6}}{\sqrt[3]{2}}`

• ^b√a=a^1/b

`\\ \rm\Rrightarrow \frac{6^{(1)/(4)}}{2^{(1)/(3)}}`

• 6=2×3

`\\ \rm\Rrightarrow \frac{2^{(1)/(4)}3^{(1)/(4)}}{2^{(1)/(3)}}`

• a^m÷a^n=a^m-n

`\\ \rm\Rrightarrow 2^{(1)/(4)-(1)/(3)}3^{(1)/(4)}`

`\\ \rm\Rrightarrow 2^{(-1)/(12)}3^{(1)/(4)}`

`\\ \rm\Rrightarrow \frac{3^{(1)/(4)}}{2^{(1)/(12)}}`

• Equalise exponential denominators

`\\ \rm\Rrightarrow \frac{3^{(3)/(12)}}{2^{(1)/(12)}}`

`\\ \rm\Rrightarrow \left((3^3)/(2)\right)^{(1)/(12)}`

`\\ \rm\Rrightarrow \sqrt[12]{(3^3)/(2)}`

`\\ \rm\Rrightarrow \sqrt[12]{(3^3* 2^82^3)/(22^82^3)}`

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

`\\ \rm\Rrightarrow \frac{\sqrt[12]{6912(8)}}{2}`

`\\ \rm\Rrightarrow \frac{\sqrt[12]{55296}}{2}`

Answer 2

`\frac{\sqrt[12]{55296} }{2}`

Explanation:

Given expression:

`\frac{\sqrt[4]{6}}{\sqrt[3]{2}}`

`\textsf{Apply exponent rule} \quad \sqrt[n]{a}=a^{(1)/(n)}:`

`\implies \frac{\sqrt[4]{6}}{\sqrt[3]{2}}=\frac{6^{(1)/(4)}}{2^{(1)/(3)}}`

Multiply the numerator and denominator by
`2^{(2)/(3)}` :

`\implies \frac{6^{(1)/(4)}}{2^{(1)/(3)}} * \frac{2^{(2)/(3)}}{2^{(2)/(3)}}`

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

`\implies \frac{6^{(1)/(4)}}{2^{(1)/(3)}} * \frac{2^{(2)/(3)}}{2^{(2)/(3)}}=\frac{6^{(1)/(4)} \cdot 2^{(2)/(3)}}{2^{(1)/(3)+(2)/(3)}}=\frac{6^{(1)/(4)} \cdot 2^{(2)/(3)}}{2}`

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

`\implies \frac{6^{(1)/(4)} \cdot 2^{(2)/(3)}}{2}=\frac{6^{(3)/(12)} \cdot 2^{(8)/(12)}}{2}`

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

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

Simplify the operation in the parentheses:

`\implies ((6^3 \cdot 2^(8))^(1)/(12))/(2)=((216\cdot 256)^(1)/(12))/(2)=((55296)^(1)/(12))/(2)`

`\textsf{Finally, apply exponent rule} \quad a^{(1)/(n)}=\sqrt[n]{a}:`

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