# If 27sqrt(3)=sqrt(3)xx3^k, what is k?

Jan 19, 2017

Write each value in the expression as a power of three, and you will see the answer is $k = 3$

#### Explanation:

Since $\sqrt{3} = {3}^{\frac{1}{2}}$ and $27 = {3}^{3}$ we can write the expression as

$\frac{{3}^{3} \cdot {3}^{\frac{1}{2}}}{3} ^ \left(\frac{1}{2}\right) = {3}^{k}$

Cancel both ${3}^{\frac{1}{2}}$ powers to get

${3}^{3} = {3}^{k}$

Jan 19, 2017

$k = 3$

#### Explanation:

Note that $27 = {3}^{3}$, so we find:

${3}^{k} = \frac{27 \textcolor{red}{\cancel{\textcolor{b l a c k}{\sqrt{3}}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\sqrt{3}}}}} = 27 = {3}^{3}$

Hence $k = 3$

Jan 19, 2017

$k = 3$

#### Explanation:

$\frac{27 \sqrt{3}}{\sqrt{3}} = {3}^{k}$

i.e. ${3}^{k} = \frac{27 \cancel{\sqrt{3}}}{\cancel{\sqrt{3}}} = 27 = 3 \times 3 \times 3 = {3}^{3}$

And thus $k = 3$.

If this were not a perfect cube, we could take $\text{logs}$ of both sides:

$k \log 3 = 3 \log 3$