# Evaluate the following?

Oct 27, 2016

${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = \frac{7}{11}$

#### Explanation:

Using the rules of exponents that ${\left(\frac{a}{b}\right)}^{x} = {a}^{x} / {b}^{x}$, and that ${\left({a}^{x}\right)}^{y} = {a}^{x y}$, we have

${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = {117649}^{\frac{1}{6}} / {1771561}^{\frac{1}{6}}$

$= {\left({7}^{6}\right)}^{\frac{1}{6}} / {\left({11}^{6}\right)}^{\frac{1}{6}}$

$= {7}^{6 \cdot \frac{1}{6}} / {11}^{6 \cdot \frac{1}{6}}$

$= {7}^{1} / {11}^{1}$

$= \frac{7}{11}$

Oct 27, 2016

$\frac{7}{11}$

#### Explanation:

This is exactly the same concept as a previous question with an index of $\frac{1}{8}$

$\rightarrow$ find the prime factors.
$\rightarrow$ index of $\frac{1}{6}$ means the same as $\sqrt[6]{\textcolor{w h i t e}{n}}$

$\frac{117649}{1771561} = \frac{7 \times 7 \times 7 \times 7 \times 7 \times 7}{11 \times 11 \times 11 \times 11 \times 11 \times 11}$

${\left(\frac{{7}^{6}}{{11}^{6}}\right)}^{\frac{1}{6}} = \sqrt[6]{\frac{{7}^{6}}{{11}^{6}}}$

=$\frac{7}{11}$

Oct 27, 2016

${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = \frac{7}{11}$

#### Explanation:

Check $117649$ for its prime factorisation:

• Last digit is odd, so not divisible by $2$.
• Sum of digits is $1 + 1 + 7 + 6 + 4 + 9 = 28$ not divisible by $3$.
• Last digit is not $0$ or $5$, so not divisible by $5$
• $\frac{117649}{7} = 16807$, so divisible by $7$
• $\frac{16807}{7} = 2401$
• $\frac{2401}{7} = 343$
• $\frac{343}{7} = 49$
• $\frac{49}{7} = 7$

So $117649 = {7}^{6}$

So if the answer is exact, we require the denominator to be a perfect $6$th power too.

Let's do some approximating...

$\frac{1771561}{117649} \approx \frac{177}{12} \approx 14$

$\sqrt{14} \approx 4$

$\sqrt[3]{4} \approx \frac{3}{2}$

So: ${\left(14\right)}^{\frac{1}{6}} \approx \frac{3}{2}$

$\frac{3}{2} \cdot 7 \approx 11$

Try:

• $\frac{1771561}{11} = 161051$
• $\frac{161051}{11} = 14641$
• $\frac{14641}{11} = 1331$
• $\frac{1331}{11} = 121$
• $\frac{121}{11} = 11$

So $1771561 = {11}^{6}$

Hence:

${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = {\left({7}^{6} / {11}^{6}\right)}^{\frac{1}{6}} = \frac{7}{11}$

Oct 27, 2016

$\frac{7}{11}$

#### Explanation:

Computing the rational number with fractional exponent ${\left(\frac{u}{v}\right)}^{\frac{1}{6}} = {u}^{\frac{1}{6}} / {v}^{\frac{1}{6}}$ is determined by prime factorizing the numerator and denominator .

example of: sqrtx = x^(1/2:

$\sqrt{\left(\frac{4}{9}\right)} = {\left(\frac{4}{9}\right)}^{\textcolor{b l u e}{\frac{1}{2}}} = {\left({2}^{2}\right)}^{\textcolor{b l u e}{\frac{1}{2}}} / {\left({3}^{2}\right)}^{\textcolor{b l u e}{\frac{1}{2}}} = \frac{2}{3}$

The rational number above ${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = {\left(117649\right)}^{\frac{1}{6}} / {\left(1771561\right)}^{\frac{1}{6}}$

Prime Factorization :
$\textcolor{b l u e}{117649 = 7 \times 7 \times 7 \times 7 \times 7 \times 7 = {7}^{6}}$
$\textcolor{b r o w n}{1771561 = 11 \times 11 \times 11 \times 11 \times 11 \times 11 = {11}^{6}}$

${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = {\left(117649\right)}^{\frac{1}{6}} / {\left(1771561\right)}^{\frac{1}{6}} = {\left(\textcolor{b l u e}{{7}^{6}}\right)}^{\frac{1}{6}} / {\left(\textcolor{b r o w n}{{11}^{6}}\right)}^{\frac{1}{6}}$

Then we apply the power of a power with base $a$ :
${\left({a}^{m}\right)}^{\frac{1}{n}} = {a}^{\frac{m}{n}}$

${\left(\frac{117649}{1771561}\right)}^{\frac{1}{6}} = \left({7}^{\frac{6}{6}} / {11}^{\frac{6}{6}}\right) = \frac{7}{11}$

Oct 27, 2016

Calculator reveals the value as

0.6363636363..

$= \left({10}^{- 2} / \left(1 - {10}^{- 2}\right)\right) 63 = \frac{63}{99} = \frac{7}{11}$