Simplify the radical expression

Question

Simplify the following: $$\sqrt{48 a^2 b^7}$$

All I have so far is 16 I think i forgot how to do this, can anyone help?

Answer 1

Try using the fact that $\sqrt{x\cdot y}=\sqrt{x}\cdot\sqrt {y}$. For example, $$\sqrt{72}=\sqrt{36\cdot 2}=\sqrt{36}\cdot\sqrt{2}=6\sqrt{2}$$ Can you see how this would relate to your example?

Answer 2

$$\begin{align}48&=\underbrace{2\times 2}\times\underbrace{2\times2}\times3\\a^2&=\underbrace{a\times a}\\b^7&=\underbrace{b\times b}\times \underbrace{b\times b}\times \underbrace{b\times b}\times b\end{align}$$

When you take the square root of them:

$$\begin{align}\sqrt{48}&=\underbrace{\sqrt{2\times 2}}_2\times\underbrace{\sqrt{2\times2}}_2\times\sqrt3\\a^2&=\underbrace{\sqrt{a\times a}}_{a}\\b^7&=\underbrace{\sqrt{b\times b}}_{b}\times \underbrace{\sqrt{b\times b}}_{b}\times \underbrace{\sqrt{b\times b}}_{b}\times \sqrt{b}\end{align}$$

So you're left with:

$$4ab^3\sqrt{3b}$$

Note that: $$(4ab^3)^\frac22\sqrt{3b}=\sqrt{16a^2b^6}\sqrt{3b}=\sqrt{48a^2b^7}$$