# How do you evaluate (56)^(7/2)?

Oct 7, 2016

$351232 \sqrt{14} \text{ }$ as an exact value

$1314189.807 \text{ }$ to 3 decimal places as an approximate value

#### Explanation:

First of all lets do this using logs:

Let $x = {\left(56\right)}^{\frac{7}{2}}$

$\text{ "log(x)" "=" "log[ (56)^(7/2)]" " =" } \frac{7}{2} \log \left(56\right) \approx 6.1186 \ldots$

$\implies \textcolor{g r e e n}{x = {\log}^{- 1} \left(6.1186 . .\right) \approx 1314189.807}$ to 3 decimal places

$\textcolor{b l u e}{\text{This is not a precise solution but will do as a check.}}$
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Consider again $x = {\left(56\right)}^{\frac{7}{2}}$

This is the same as: $\sqrt{{56}^{7}} \text{ "=" } \sqrt{{56}^{2} \times {56}^{2} \times {56}^{2} \times 56}$

$= {56}^{3} \sqrt{56}$

Splitting 56 into a product of primes we observe:

Giving:

${56}^{3} \sqrt{{2}^{2} \times 14}$

$2 \times {56}^{3} \sqrt{14} \textcolor{g r e e n}{\approx 1314189.807}$ to 3 decimal places

$\textcolor{b l u e}{\text{So the exact value is: } 351232 \sqrt{14}}$