Simplify Square Root Expression $\sqrt{125} - \sqrt{5}$

Question

$\sqrt{125}-\sqrt5$ simplify it.

I thought it would be $\sqrt {5\cdot5\cdot5}-\sqrt 5$ which would be the square root of 25 which is 5 but it is not.

Can you show how to simplify this?

Answer 1

$\sqrt{125}-\sqrt{5}=\sqrt{5\times 25}-\sqrt{5}=\sqrt{25}\sqrt{5}-\sqrt{5}=5\sqrt{5}-\sqrt{5}=4\sqrt{5}$

Answer 2

So you are asking

$$ \sqrt{125} - \sqrt{5} $$

This is

$$ \sqrt{25 \times 5} - \sqrt{5} $$

Which simplifies to

$$ 5 \sqrt{5} - \sqrt{5} $$

Which factors to

$$ \sqrt{5}(5 - 1) $$

Thus the answer is

$$ 4 \sqrt{5} $$

Answer 3

$$(\sqrt{125}-\sqrt5)^2=125+5-2\sqrt{625}=130-50=80$$ therefore $$\sqrt{125}-\sqrt 5=\sqrt{80}=4\sqrt 5$$

It is not the most usual way to do this. Just giving an alternative solution.

Answer 4

Try thinking about it as $ax - bx$. The most obvious choice is $x = \sqrt{5}$, so that then $b = 1$. Now you just need to rewrite $\sqrt{125}$ as $a \sqrt{5}$, and it turns out that $\sqrt{125} = 5 \sqrt{5}$.

So now you just do $5 \sqrt{5} - \sqrt{5} = 4 \sqrt{5}$.

Answer 5

Starting from where you left off:

$$ \begin{eqnarray} \sqrt{125}-\sqrt5 &=& \sqrt {5\cdot5\cdot5} - \sqrt 5 \\ &=& \sqrt{5} \cdot \sqrt{5} \cdot\sqrt{5}-\sqrt 5 \\ &=& \sqrt{5} (\sqrt{5} \cdot\sqrt{5} - 1) \\ &=& \sqrt{5} (5 - 1) \\ &=& 4 \sqrt{5} \end{eqnarray} $$

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If the question were $\sqrt{125} \div \sqrt5$, then the answer would indeed be $5$.