Simplifying square roots

▶ Simplifying square roots

Roots are nice, but we prefer dealing with regular numbers as much as possible. So, for example, instead of √4 we prefer dealing with 2. What about roots that aren't equal to an integer, like √20? Still, we can write 20 as 4⋅5 and then use known properties to write √(4⋅5) as √4⋅√5, which is 2√5. We *simplified* √20.

⏱ 3:09 💾 1.8 MB License: cc-by-nc-sa Streaming via CDN

▶ Simplifying square roots (variables)

A worked example of simplifying radical with a variable in it. In this example, we simplify 3√(500x³).

⏱ 4:16 💾 3.2 MB License: cc-by-nc-sa Streaming via CDN

▶ Simplifying square-root expressions

Worked examples of taking expressions with square roots and taking all of the perfect squares out of the square roots. For example, 2√(7x)⋅3√(14x²) can be written as 42x√(2x).

⏱ 8:30 💾 6.5 MB License: cc-by-nc-sa Streaming via CDN

📄 Simplifying square roots review

Example

Let's simplify $\sqrt{75}$ by removing all perfect squares from inside the square root.

We start by factoring $75$, looking for a perfect square:

$75=5\times5\times3=\blueD{5^2}\times3$.

We found one! This allows us to simplify the radical:

$\begin{align}
\sqrt{75}&=\sqrt{\blueD{5^2}\cdot3}
\\
&=\sqrt{\blueD{5^2}} \cdot \sqrt{{3}}
\\
&=5\cdot \sqrt{3}
\end{align}$

So $\sqrt{75}=5\sqrt{3}$.

Want another example like this? Check out this video.

Practice

[[☃ graded-group-set 2]]

Want to try more problems like these? Check out this exercise.

Simplifying square roots with variables

Example

Let's simplify $\sqrt{54x^7}$ by removing all perfect squares from inside the square root.

First, we factor $54$:

$54=3\cdot 3\cdot 3\cdot 2=3^2\cdot 6$

Then, we find the greatest perfect square in $x^7$:

$x^7=\left(x^3\right)^2\cdot x$

And now we can simplify:

$\begin{align}
\sqrt{54x^7}&=\sqrt{3^2\cdot 6\cdot\left(x^3\right)^2\cdot x}
\\
&=\sqrt{3^2}\cdot \sqrt6 \cdot\sqrt{\left(x^3\right)^2}\cdot \sqrt x
\\
&=3\cdot\sqrt6\cdot x^3\cdot\sqrt x
\\
&=3x^3\sqrt{6x}
\end{align}$

Practice

[[☃ graded-group-set 2]]

Want to try more problems like these? Check out this exercise.

More challenging square root expressions

[[☃ graded-group-set 3]]

Want to try more problems like these? Check out this exercise.