Multiplying Radicals BlackBoard

Multiplying Radicals – The Complete Lesson with Recap

In this lesson, you will learn all about multiplying radicals and their different types.

Don’t worry if you don’t know what a radical is, you have probably seen them before! They are numbers containing the symbol, which is called the radical or root symbol.

Radicals are a tidy way to write numbers that have decimal expansions that go on endlessly, so you must be able to use them confidently.

Multiplying radicals is often better seen as simplifying a radical expression, which is rewriting it as simply as possible!

But first, let’s get comfortable with using the correct words.

Contents

Parts of a Radical and Square Roots
Multiplying Radicals
Method 1: Using Radical Notation
Method 2: Using Fractional Exponents
Which Method to Use
How to Simplify Radical Expressions
Recap
Why Use a Radical?
Square Roots
Cube Roots and More
To Sum Up (Pun Intended)

Parts of a Radical and Square Roots

Parts of Radical Expressions

The radicand, a+3, sits inside the radical symbol, . The radical symbol just helps you see clearly what’s going on!

A square root is a type of radical that you’re probably familiar with!

3 Ways of Showing Square Roots

The exponent for a square root is left out – if you ever see a radical symbol without a number for its root, then it is a square root.

If you are not sure about how radicals work or want to brush up on it before tackling the multiplication, skip to the Recap section and come back here when you’re done.

Multiplying Radicals

If you prefer reading to watching videos, or want to do both, then here we go!

Now that you know what the different types and parts of radical expressions are and how to simplify them, you are ready to start multiplying them together!

There are two useful methods for multiplying radicals:

  1. Radical Notation, for equations with the same exponents, like \(\left(\sqrt{2}\times\sqrt{6}\right)\) or \(\left(\sqrt[5]{9}\times\sqrt[5]{12}\right)\)
  2. Fractional exponents, for radicals with the same radicand, like \(\left(\sqrt[5]{387}^4\times\sqrt[3]{387}^6\right)\) or \(\left(\sqrt{10}\times\sqrt[4]{10}^3\right)\)

Method 1: Using Radical Notation

There are a few simple rules that help when multiplying one radical expression with another. We’ll go through them one at a time.

Rule 1: The radicands multiply together and stay inside the radical symbol.

Radicands Multiply Inside

\(
\sqrt{5}\times\sqrt{2}=\sqrt{10}
\)

\(
\sqrt{8}\times\sqrt{2}=\sqrt{16}
\)

But 16 is a nice, square number, so this can be simplified.

\(
\sqrt{16}=4
\)

Because 42 = 4 × 4 = 16.

Rule 2: Any numbers outside the radical symbol multiply together and stay outside the radical symbol.

Numbers Outside Radical Multiply

\(
3\sqrt{5}\times2\sqrt{2}=6\sqrt{10}
\)

\(
5\sqrt{8}\times2\sqrt{2}=10\sqrt{16}
\)

Again, 16 is a square number, so this can also be simplified.

\(
\eqalign{
10\sqrt{16}&=10\times4\\
&=40
}
\)

Rule 3: You must apply the radical to a number to take it outside the radicand.

Take a Square Outside the Radical

See how that works for a square root? The a2 could be removed because its power was 2, the inverse of the root, but the power of b3 doesn’t match the root so it stays inside.

The same applies to cubes or any other order, for that matter.

Take a Cube Outside the Radical

In this case, a2 stays inside because it is not a cube number, while b3 was, so it can be brought out.

\(
\eqalign{
\sqrt{50}&=\sqrt{25\times2}\\
&=\sqrt{25}\times\sqrt{2}
}
\)

And the square root of 25 is 5, because 52 = 5 × 5 = 25. So…

\(
\sqrt{25}\times\sqrt{2}=5\sqrt{2}
\)

This is as simple as this radical expression gets.

\(
\sqrt[3]{108}=\sqrt[3]{27}\times\sqrt[3]{4}
\)

The cube root of 27 is 3, because 33 = 3 × 3 × 3 = 27.

\(
\sqrt[3]{27}\times\sqrt[3]{4}=3\sqrt[3]{4}
\)

Method 2: Using Fractional Exponents

First off, what on earth is a fractional exponent?!

Parts of a Fraction Exponent

Fractional means a fraction shows it with a numerator on the top (multiplier) and a denominator on the bottom (divisor).

Exponent is another word for power, or index.

Put them together, and you have a fractional exponent – a handy way of showing a radical expression because it allows you to see the powers or roots as simple fractions.

The rule to know when using this method is one of the exponent product rules.

Exponent Product Rule Base Unchanged

With this power-ful rule, you now have the keys to multiplying radicals of all sorts!

Here is a worked example.

Fractional Exponent Multiplication Worked Example

Which Method to Use When Multiplying Radicals?

Both methods will work either way once you understand what they do, but which one is easiest to use?

The answer is, whichever one you prefer.

However!! This quick check will give you the least work to do:

  • If the exponents (roots/indices/powers) are the same, then the radical method is the simplest.
  • If the radicands (bases) are the same, then the fractional exponent method is simplest

That’s all there is to it, now…

How to Simplify Radical Expressions

Simplifying a radical is a must-do step for presenting your final answer. To not do this would be like showing 2/4 instead of 1/2.

It’s the reverse of the radical multiplication rule. Let’s try it out together.

75 is not a square number, so what can you do to simplify it?

75 can be factorized into 25 and 3, because 25 × 3 = 75. You may have spotted that 25 is a square number!

So, if we use the first rule of multiplying radicals in reverse:

\(
\sqrt{75}=\sqrt{25}\times\sqrt{3}
\)

\(\sqrt{3}\) can’t be simplified, but \(\sqrt{25}=5\) so \(\sqrt{75}\) can be simplified to

\(
\sqrt{75}=5\sqrt{3}
\)

Let’s do one more together before you do it yourself, this time with cube roots.

First, we must look for a cube number that divides 250.

You can look up the cube numbers; the first few are 8, 27, 64, 125, 196,…

250 is divisible by 125, and 125 is a cube number. Perfect!

So, again using the first rule of multiplying radicals:

\(
\eqalign{
\sqrt[3]{250}&=\sqrt[3]{125\times2}\\
&=\sqrt[3]{125}\times\sqrt[3]{2}
}
\)

Now \(\sqrt[3]{125}=5\) and \(\sqrt[3]{2}\) can’t be simplified, so:

\(
\sqrt[3]{125}\times\sqrt[3]{2}=5\sqrt[3]{2}
\)

To spot square factors that can help simplify radicals, use our list of factors for all numbers up to 100.

Recap

Why Use a Radical?

The answer is: look at the square root of any number other than a square number, like 7.

\(
\sqrt{7}=2.64575131\ldots
\)

That decimal number goes on… and on… and on… FOREVER!

Numbers like \(\sqrt{7}\) are known as surds if they use the radical notation, or irrational numbers otherwise. The root or radical symbol helps us write them easily, so that equations aren’t full of decimal numbers.

Square Roots

If a is a positive number, a square root of a is a number that gives a when squared, or multiplied by itself. Just like this:

\(
\sqrt{25}=5
\)

Because 5 multiplied by itself is 25.

What happens if you square a square root?

\(
\left(\sqrt{5}\right)^2=\left(5^\left(\frac{1}{2}\right)\right)^2
\)

And when squaring this, it’s the same as multiplying the exponent by 2:

\(
\left(5^\left(\frac{1}{2}\right)\right)^2=5^\left(\frac{2}{2}\right)=5^1
\)

So, the square root does the opposite of squaring a number.

Impress your teacher by telling them that a square root is the inverse of a square function. Just make sure you can explain what that means!

Cube Roots and More

A cube root asks, “What number would multiply with itself 3 times to make the radicand?”

It looks like this:

8's Cube Root

As you can see, the answer for the cube root of 8 is 2, because

23 = 2 × 2 × 2 = 8

A 4th root looks like \(\sqrt[4]{16}\)

It asks, “What number would multiply with itself 4 times to make the radicand?”

The answer to \(\sqrt[4]{16}\) is 2, because 24=2×2×2×2=16

Now go learn what you came here for in the first place. Head back to the “how-to” section on Multiplying Radicals

To Sum Up (Pun Intended)

The radical symbol is another way of writing a fractional exponent a12

The radicand is the number inside the radical symbol.

Radicals with root x ask the question: “What would multiply itself x number of times to make the radicand.”

To simplify a radical expression, look for a number that divides the radicand. Use our table of factors and multiples to help with this.

When multiplying radicals:

  • If they have the same exponent (root), use the radical method. Numbers outside the radical multiply and stay outside. Numbers inside the radicands multiply and stay inside the radical.
  • If they have the same radicand, convert to fractional exponents. Exponents add together, and the radicand, which is now called the base, stays the same.

If any of this didn’t make sense, go back over it in the lesson above and make sure you can answer all the practice questions. If you’re stuck, comment below, and we can help out!

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