Multiplying radicals (advanced)

What Are Radicals?

Before we get into the actual mathematics behind radicals, let’s first define what we mean by the term “radical”. Simply put, a radical is some number, which we call the radicand, that is held within a root – that is, a square root, cube root, etc. These roots are also sometimes referred to as the radical sign. Here are a few examples:

10\sqrt{10}
2x\sqrt{2x}
10057xyz4\sqrt{10057xyz^4}

It’s also important to note that anything, including variables, can be in the radicand!

As you progress in mathematics, you will commonly run into radicals. Thus, it is very important to know how to do operations with them. In this article, we will look at the math behind simplifying radicals and multiplying radicals, also sometimes referred to as simplifying and multiplying square roots. In order to have a better grip on the concepts in this lesson, reviewing the basic on simplifying radicals, and adding and subtracting radicals is recommended.

Multiplying Radicals

Multiplying radicals, though seemingly intimidating, is an incredibly simple process! Before we get into multiplying radicals directly, however, it is important to review how to simplify radicals.

In order to simplify a radical, all we need to do is take the terms of the radicand out of the root, if it’s possible. Let’s look at three examples:

Example 1:

Simplify 9\sqrt{9}

This example should be very straightforward. All we need to do is take the square root of 9!

Answer: 3

Example 2:

Simplify 169x4\sqrt{169x^4}

This example is a little more difficult, but nonetheless is simple when we break it down. To simplify more complex radicals, it is often helpful to break the radicand down and simplify individual terms. At least at first until you get the hand of it!

169×x4\sqrt{169 \times x^4}

Now that our radicand is broken down, let’s take the square root of both terms and solve!

Answer: 13x213x^2

Example 3:

Simplify 10x4\sqrt{10x^4}

This example is very similar to the previous example, but is a little different after with break the radicand down and try to solve.

10×x4\sqrt{10 \times x^4}

You should notice at this point that there is no integer square root of 10. Therefore, we simply just leave it as a radical, and only simplify x4x^4.

Answer: x210x^2\sqrt{10}

Now that we know how to simplify radicals, let’s briefly look at how to multiply radicals and multiply square roots before doing some example problems.

To multiply radicals, if you follow these two rules, you’ll never have any difficulties:

1) Multiply the radicands, and keep the answer inside the root

2) If possible, either before or after multiplication, simplify the radical.

Now that we know what we mean by “multiplying radicals”, let’s look at the process behind the work and actually multiply radicals in some example problems.

How To Multiply Radicals

The best way to learn how to multiply radicals and how to multiply square roots is to practice with some more sample problems.

Example 1:

Solve 6×2\sqrt{6} \times \sqrt{2}

In this example, we first need to multiply the radicands of each radical

12\sqrt{12}

And that’s it! We can’t simplify this radical, as there is no integer square root of 12, so therefore this is our final answer.

Example 2:

Solve 5x×5x\sqrt{5x} \times \sqrt{5x}

This example involves some variables, but is still very simple to solve. First, let’s multiply the radicands before seeing if we can simplify anything.

25x2\sqrt{25x^2}

Now that we’ve done our multiplication, you should notice that we can simplify this radical by taking the square root of 25 and of x2x^2. This gives us our final answer of:

5x5x

Example 3:

Solve 320rt×36qr2{^3}\sqrt{20rt} \times {^3}\sqrt{6qr^2}

Don’t be intimidated by this example! Even though we’re dealing with cube roots instead of multiplying square roots, our process doesn’t change. First, let’s multiply the radicands.

3120r3tq{^3}\sqrt{120r^3tq}

Now, let’s look at each individual term and see if we can simplify anything. Hopefully you’ll notice there is only one term that we can take the cube root of, r3r^3. The rest simply just stays inside the radical and we have our final answer!

r3120tqr{^3}\sqrt{120tq}

Example 4:

Solve 2xyz×11×3y3\sqrt{2xyz} \times \sqrt{11} \times 3\sqrt{y^3}

Don’t be intimidated by this example either! The process is still the exact same thing as we’ve been doing. See that 3 in front of the last radical? Just leave it alone. It doesn’t get multiplied. Now let’s multiply all three of these radicals.

322xy4z3\sqrt{22xy^4z}

Now let’s see if we can simplify this radical any more. You should notice that we can only take out y4y^4 from the radicand. This gives us our final answer of:

3y222xz3y^2\sqrt{22xz}

Example 5:

Solve 32×3{^3}\sqrt{2} \times \sqrt{3}

This example is actually more of a trick question. Since the roots we are multiplying are not the same, and there is no simplification we can do right now, we actually can’t go any further with our answer! Don’t worry too much about multiplying radicals with different roots. These questions are very uncommon and oftentimes there is little to be done to solve them without the help of calculators.

And that’s all there is to it! We can now successfully multiply any given radicals! The work with radicals doesn’t stop here, however. Click on the following links for further work with radicals in basic radical functions, transformations of functions, and solving radical equations. As well, for more practice, take a look at the lesson on dividing radicals!

Multiplying radicals (advanced)

It requires 2 steps to multiply radicals. First is to multiply the numbers inside the radical sign, the radicands, together. Second is to multiply the numbers outside the radical sign together. Don't forget that only radicals with the same index can be combined through multiplication!

Lessons

  • 2.
    Multiplying Radicals
    Multiply and simplify.
    • b)
      (28)(518)\left( {2\sqrt 8 } \right)\left( {5\sqrt {18} } \right)
    • c)
      (5x)(6y)\left( {5\sqrt x } \right)\left( {6\sqrt y } \right)
    • d)
      33\sqrt {3} \cdot \sqrt {3}
  • 3.
    Expand and simplify.
    • a)
      3(563)\sqrt 3 \;\left( {5\sqrt 6 - \sqrt 3 } \right)
    • c)
      3(27)3(1221)3\;\left( {\sqrt 2 - \sqrt 7 } \right) - \sqrt 3 \left( {\sqrt {12} -\sqrt {21} } \right)
    • d)
      3x(x5y)-3\sqrt x\;\left( {\sqrt x -5 \sqrt y } \right)
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Multiplying radicals (advanced)

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