Rationalising denominator with cubed surd

[aajlh]

I need to rationalise the denominator of a cube root. I have only done this before with a
square root as the denominator.

The question is:

1 / cube root of 5.


Firstly, I am not entirely sure how to get rid of the cube root surd in the denominator. I tried in
a way that I think I am rationalising the denominator correctly but then I get an answer
different from the solution in the text.

Secondly, I also don't know how to go from the answer I got which is: (cube root 5)^3 / 5 to the
answer in the book having the numerator to a power, ie: 5^2/3 / 5.

The description below should explain what I mean.

I have only rationalised the denominator before using square roots. I assume that in this
case I should multiply the numerator and denominator by something to get rid of the
cube root of 5 in the denominator but just multiplying them both by the cube root of 5
wouldn't work as this would only square it. So perhaps I should multiply the numerator and
denominator by (cube root 5)^3?

ie, 1 / cube root of 5 * (cube root 5)^3 / (cube root 5)^3.

That gives me an answer of (cube root five)^3 / 5 (which is good because I got rid of the surd in the denominator).

The problem is the answer in the book is 5 ^ 2/3 all over 5.

What have I done wrong? Did I multiply the numerator and denominator by the right thing? How did they end up with the answer in the text having the numerator to the powerof 2/3?

Any thought would be appreciated.

Thanks

Anne

 

[galactus]

Remember, when we multiply exponents, we add the exponents.

So, 1/3+x=1

Multiply top and bottom by \(\displaystyle \sqrt[3]{5^{2}}=5^{\frac{2}{3}}\)

\(\displaystyle \frac{1}{5^{\frac{1}{3}}}\cdot \frac{5^{\frac{2}{3}}}{5^{\frac{2}{3}}}\)

The denominator becomes 5 and we get:

\(\displaystyle \frac{5^{\frac{2}{3}}}{5}=\frac{\sqrt[3]{5^{2}}}{5}=\frac{\sqrt[3]{25}}{5}\)

Your answer of \(\displaystyle \frac{(\sqrt[3]{5})^{3}}{5}=1\)

We do the same thing when rationalizing a square root. It's just that 1/2+1/2=1, so we just multiply by the same exponent.