Question
Prove that:
$\bigg[7\Big\{(81)^\frac{1}{4}+(256)^\frac{1}{4}\Big\}^\frac{1}{4}\bigg]^4=16807$
$\bigg[7\Big\{(81)^\frac{1}{4}+(256)^\frac{1}{4}\Big\}^\frac{1}{4}\bigg]^4=16807$
✓
Answer
$\text{L.H.S}=\bigg[7\Big\{(81)^\frac{1}{4}+(256)^\frac{1}{4}\Big\}^\frac{1}{4}\bigg]^4=16807$
$=\bigg[7\Big\{3^{4\times\frac{1}{4}}+4^{4\times\frac{1}{4}}\Big\}^\frac{1}{4}\bigg]^4$
$=\bigg[7\big\{3+4\big\}^\frac{1}{4}\bigg]^4$
$=\Big[7\big\{7\big\}^\frac{1}{4}\Big]^4$
$=7^4\times7^{\frac{1}{4}\times4}$
$=7^4\times7$
$=7^5$
$=16807$
$=\text{R.H.S.}$
$=\bigg[7\Big\{3^{4\times\frac{1}{4}}+4^{4\times\frac{1}{4}}\Big\}^\frac{1}{4}\bigg]^4$
$=\bigg[7\big\{3+4\big\}^\frac{1}{4}\bigg]^4$
$=\Big[7\big\{7\big\}^\frac{1}{4}\Big]^4$
$=7^4\times7^{\frac{1}{4}\times4}$
$=7^4\times7$
$=7^5$
$=16807$
$=\text{R.H.S.}$
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