# Multiplying variables with different bases and different exponents

I'm stuck. Helping kid with alg II and the instructions say to simplify the expression into one radical. $$\sqrt{10} \cdot \sqrt{3}$$ I know how to do it with same base, or same exponent, but ten to the 1/2 times 3 to the 1/4? Am I just overthinking this? I can't see it.

• First step, $(\sqrt{100})(\sqrt{3})$. In general, if dealing with $x^{a/b}y^{c/d}$, first step is to bring $a/b$ and $c/d$ to a common denominator. – André Nicolas Dec 17 '15 at 20:21

$$10^{\frac{1}{2}} 3^{\frac{1}{4}} = 10^{\frac{2}{4}} 3^{\frac{1}{4}} = (10^2 \times 3)^{\frac{1}{4}} = 300^{\frac{1}{4}}$$
This method will work for $x^ay^b$ when $a$ and $b$ are rational - just write the exponents using their least common denominator.
The trick is to realize that $\sqrt{10}$ can be rewritten as $\sqrt{100}$. There are at least two ways to make this connection:
• You could recognize that $10 = \sqrt{100}$, so $\sqrt{10}=\sqrt{\sqrt{100}}$.
• You could write $\sqrt{10}$ as $10^{1/2}$, and then rewrite $1/2$ as $2/4$, so we have $10^{2/4} = (10^2)^{1/4}$.