Conquering Radicals: Your Essential Guide to Simplifying Square Roots in Algebra 1

Once you know how to use exponents in Algebra 1, you need to learn how to use their opposite, radicals (square roots). It is very important to know how to work with radicals, especially when you need to simplify the messy results that the Quadratic Formula gives you.

But students often have trouble with the rules for adding, multiplying, and, most confusingly, simplifying square roots, which seem to be random.

This mathaversity guide gives you a clear, step-by-step plan for mastering radicals. It focuses on the three skills you need to do well in this unit. And to help you remember what you've learned, we'll show you where to get free practice worksheets from our website right now!

The Most Important Skill: Making Radicals Easier

You can't always find the value of a square root, like $\sqrt{5}$ , but you can always make it easier to understand.

You can't always make it easier, but you can always do it. To simplify, you need to take out all the factors that are perfect squares from under the radical sign $(\sqrt )$ , which leaves the smallest number inside.

🫵 Step-by-Step for Making x Easier $\sqrt{x}$

List of Perfect Squares: Write down or memorize the first few perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100
Find the largest factor: Find the biggest perfect square that can be divided into the number under the radical (the radicand).
Factor the Radicand: Write the radicand as the product of the perfect square and the other factor.
To simplify, find the square root of the perfect square factor and put it outside the radical. Keep the factor that isn't a perfect square inside.

For example, Simplify $\sqrt{72}$
1. Find the factor: 36 is the biggest perfect square that 72 can be divided by.
2. Factor: $\sqrt{72}=\sqrt{36.2}$
3. Simplify: $\sqrt{36} .\sqrt{2}=6\sqrt{2}$

🫵 The Rules of the Game for Working with Radicals

The rules for adding and multiplying radicals are very different. A common mistake is to mix them up.

The Easy Part: Multiplying and Dividing Radicals

If two radicals have the same index (like both are square roots), you can multiply or divide them.

Rule: Multiply (or divide) the numbers that are outside the radical and the numbers that are inside the radical. Always make the result easier to understand.

Example of Multiplication: $(3\sqrt{5}) · (2\sqrt{10})$

= (3 \cdot 2)\sqrt{5 \cdot 10} = 6\sqrt{50}

Simplify $6\sqrt{50}$ → $6\sqrt{25 \cdot 2}$ = $6 \cdot 5\sqrt{2}$ = $30\sqrt{2}$

The Strict Rule for Adding and Subtracting Radicals

Adding and subtracting radicals is similar to adding terms that are the same in a polynomial. You can only add or subtract terms that have the same radicand (the number under the

\sqrt{}

sign).

If the radicands are the same, you add the numbers outside the radical. You can't mix them together if they are different.

Example: $4\sqrt{3} + 7\sqrt{3}=11\sqrt{3}$

Example: $5\sqrt{2} + 2\sqrt{5}=$ Can't be made any simpler.

Important Strategy: If the radicands are not the same, you need to simplify each radical first! A lot of the time, simplifying will show you like terms that were hidden.

Example: $\sqrt{8} + \sqrt{18}$

Simplify:

2\sqrt{2} + 3\sqrt{2}=5\sqrt{2}

The Rule of Rationalizing the Denominator

It is common in math for a final answer to not have a radical in the bottom of a fraction. Rationalizing the denominator is the process of getting rid of it.

Step-by-Step for Rationalizing $\frac{a}{\sqrt{b}}$
Identify: You have a radical in the bottom (for example, $\frac{5}{\sqrt{3}}$ ).
Multiply the whole fraction by a special form of 1: $\frac{b}{\sqrt{b}}$
Simplify: The denominator becomes $\sqrt{b} \cdot \sqrt{b} = b$ (the radical disappears).

Example: Rationalize $\frac{5}{\sqrt{3}}$

\frac{5}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}

🫵 Using radicals to solve equations

The variable x can sometimes be stuck inside the radical. To solve these radical equations, you need to do one important thing:

Isolate the Radical: Get the $\sqrt{}$ term alone on one side of the equation.
Square both sides of the equation to get rid of the radical.
Solve the other equation: This will give you a simple linear or quadratic equation.
Check for Extra Solutions: You have to plug your final answer(s) back into the original equation to make sure they work, because squaring both sides can give you wrong answers. If they don't, they are called "extraneous solutions."

🫵 Guaranteed Success: Download Your Free Radicals Worksheets!

Knowing the rules of radicals is very important for doing well in Algebra 1, especially when you start using the Quadratic Formula. Focused repetition is the best way to remember the difference between adding and multiplying and to get good at simplifying.

🫵 Get Free Algebra 1 Worksheets to Practice Right Away

Mathaversity has high-quality, focused practice materials to help you master this subject. We have free worksheets that you can download that cover simplifying radicals, doing operations, and rationalizing the denominator. They come with step-by-step solutions!

Click here to get worksheet!

Get your free practice worksheets now and stop being confused—it's the best way to get ready for your next Algebra 1 test!

🫵 Conclusion: Simplifying Your Way to Algebra Mastery

In conclusion, the best way to master algebra is to make things easier. Radical rules are clear, but they never change. You can avoid making a lot of mistakes in Algebra 1 by always making sure your answers are simplified and rationalized and by looking for those perfect square factors.

Having trouble remembering the difference between adding and multiplying radicals?

Mathaversity offers personalized online tutoring from experts to help you understand these rules right away and feel confident about all parts of radicals and the Quadratic Formula.

Conquering Radicals: Your Essential Guide to Simplifying Square Roots in Algebra 1