exponents: solve 0.4k^(1/3) = 0.1k

can you please tell me how to get the answer to this problem which is

0.4k^(1/3) = 0.1k

the answer ends up being k = 8

art2ista said:

can you please tell me how to get the answer to this problem which is
.4k^(1/3) = .1k
the answer ends up being k=8

Click to expand...

Multiply by 10:
4k^(1/3) = k

4k^(1/3) - k = 0
k(4k^(-2/3) - 1) = 0
k = 0 or 4k^(-2/3) - 1 = 0

4k^(-2/3) = 1
1 / k^(2/3) = 1 / 4
k^(2/3) = 4
k = 4^(3/2)
k = 8

Feel better now?

\(\displaystyle \L .4k^{\frac{1}{3}} = .1k\)

multiply both sides by 10 to clear the decimals ...

\(\displaystyle \L 4k^{\frac{1}{3}} = k\)

cube both sides ...

\(\displaystyle \L 64k = k^3\)

move all terms to one side and set = 0 ...

\(\displaystyle \L 0 = k^3 - 64k\)

factor, step 1 ...

\(\displaystyle \L 0 = k(k^2 - 64)\)

factor, step 2 ...

\(\displaystyle \L 0 = k(k + 8)(k - 8)\)

as it turns out, you get three valid solutions for k ...
k = 0, k = -8, and k = 8.