# rationals

#### CSmith

##### Member
1) 125 2/3

(125 1/3)^2

=( 3 square root 125)^2
=(3 square root 5x5x5)^2
=(3 square root 5 )^2

right so far?

#### Jameson

Staff member
1) 125 2/3

(125 1/3)^2

=( 3 square root 125)^2
=(3 square root 5x5x5)^2
=(3 square root 5 )^2

right so far?
Looks correct to me. Why don't you try solving a few problems fully and post them all at once to make this more efficient? It's better to type too much than not enough with online math help.

Good job by the way on learning this stuff!

#### CSmith

##### Member
OK.

CONTINUED

3 SQUARE ROOT 5 X 3 SQUARE ROOT 5
=9(5 SQUARE ROOT 5)
= 25 square root 5

#### Jameson

Staff member
OK.

CONTINUED

3 SQUARE ROOT 5 X 3 SQUARE ROOT 5
=9(5 SQUARE ROOT 5)
= 25 square root 5
You got part of it right. The 3's will combine into 9 so now you must calculate the square root bits. After multiplying 3*3 you have $$\displaystyle \sqrt{5} \cdot \sqrt{5}$$ remaining, which is multiplied together. A common fact that is useful in these problems is that squaring and taking the square root are opposite operations, so when you do both to one number, nothing happens.

$$\displaystyle \sqrt{5} \cdot \sqrt{5}=( \sqrt{5})^2=5$$

#### Sudharaka

##### Well-known member
MHB Math Helper
1) 125 2/3

(125 1/3)^2

=( 3 square root 125)^2
=(3 square root 5x5x5)^2
=(3 square root 5 )^2

right so far?
Hi CSmith,

Note that, $$(\sqrt[3]{5\times 5\times 5})^2=5^2$$. But you have written, $$(\sqrt[3]{5\times 5\times 5})^2=(\sqrt[3]{5})^2$$ which is incorrect.

Kind Regards,
Sudharaka.

#### Sudharaka

##### Well-known member
MHB Math Helper
Hi CSmith,

I went through most of your posts and it seems to me that you are making a lot of effort to typeset the mathematics in them. I suggest you to learn a bit of LaTeX commands. It's very easy. For starters, you can read the LaTeX help section.

As an example let me show you how to typeset your question in LaTeX.

Code:
$125^{\frac{2}{3}}=(125^{\frac{1}{3}})^2$

$=(\sqrt[3]{125})^2$

$=(\sqrt[3]{5\times 5\times 5})^2$

$=(\sqrt[3]{5})^2$
This will produce,

$125^{\frac{2}{3}}=(125^{\frac{1}{3}})^2$

$=(\sqrt[3]{125})^2$

$=(\sqrt[3]{5\times 5\times 5})^2$

$=(\sqrt[3]{5})^2$

To make the equal signs align with each other you can use the "eqnarray" environment as follows,

Code:
\begin{eqnarray}

125^{\frac{2}{3}}&=&(125^{\frac{1}{3}})^2\\

&=&(\sqrt[3]{125})^2\\

&=&(\sqrt[3]{5\times 5\times 5})^2\\

&=&(\sqrt[3]{5})^2

\end{eqnarray}
will give you,

\begin{eqnarray}

125^{\frac{2}{3}}&=&(125^{\frac{1}{3}})^2\\

&=&(\sqrt[3]{125})^2\\

&=&(\sqrt[3]{5\times 5\times 5})^2\\

&=&(\sqrt[3]{5})^2

\end{eqnarray}

Kind Regards,
Sudharaka.