# 49 to the 2/3 power I have put this in many calculators and have gotten different answers?

## ${49}^{\frac{2}{3}}$

Jan 28, 2018

${49}^{\frac{2}{3}} = \sqrt[3]{2401} \approx 13.4$

#### Explanation:

${49}^{\frac{2}{3}} = {\left({7}^{2}\right)}^{\frac{2}{3}}$
${\left({7}^{2}\right)}^{\frac{2}{3}} = {7}^{2 \times \frac{2}{3}}$
${7}^{2 \times \frac{2}{3}} = {7}^{\frac{4}{3}}$
${7}^{\frac{4}{3}} = \sqrt[3]{2401}$
$\sqrt[3]{2401} \approx 13.4$

Feb 1, 2018

Ensure that you indicate to the calculator:

$\sqrt[3]{{49}^{2}} = \sqrt[3]{2401} = 13.3905$

${\left(\sqrt[3]{49}\right)}^{2} = {\left(3.6593\right)}^{2} = 13.3905$

#### Explanation:

You need to be sure to tell the calculator exactly what you mean.

This can be done using brackets, or by working out one part first, pressing the equal sign, and then doing the second part.

Some calculators have an $A N S$ key which allows you to use the previous answer in the next calculation. On simpler calculators you might need to use the Memory function.

Simply typing in: $\text{49^2" div 3}$ will be understood as ${49}^{2} / 3$ because a power is calculated before a division.

This will give the answer $800.3333333$

You might inadvertently tell the calculator ${\left(\frac{49}{3}\right)}^{2}$
This will give the answer $266.77777778$

However, if you use brackets correctly you will be showing that the base of $49$ is to be raised to an index which is a fraction.

$49 \text{^} \left(2 \div 3\right)$ will be understood as ${49}^{\frac{2}{3}}$

This actually means: $\sqrt[3]{{49}^{2}}$ or ${\left(\sqrt[3]{49}\right)}^{2}$

Both methods will give the same answer.

$\sqrt[3]{{49}^{2}} = \sqrt[3]{2401} = 13.3905$

${\left(\sqrt[3]{49}\right)}^{2} = {\left(3.6593\right)}^{2} = 13.3905$

YOu can also do two steps:

$2 \div 3 = 0.666666666 \ldots$

$49 \text{ ^ ANS} = 13.3905$

Note that the answer is irrational and has been rounded to 4 decimal places.

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If the question had been ${49}^{\frac{3}{2}}$ we would have had a rational answer:

${49}^{\frac{3}{2}} = {\left(\sqrt{49}\right)}^{3} = {7}^{3} = 343$

or

$\sqrt{{49}^{3}} = \sqrt{117649} = 343$