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Squares and Square Roots

Squares and Square Roots

At a Glance - Multiplication


Multiplication of radical expressions is similar to multiplication of polynomials. Remember what a gas that was? Now we can experience that thrill ride all over again, and you don't even need to wait in an incredibly long line first.

When multiplying radical expressions, we give the answer in simplified form. Multiplying two monomial (one-term) radical expressions is the same thing as simplifying a radical term.

Sample Problem

Multiply .

We multiply the radicands to find .

Then, we simplify our answer to .

Sample Problem

Multiply .

We distribute the and simplify the resulting terms:

Since these simplified terms have different radicands, there are no like terms to combine, so we're done. If you absolutely need a combining fix, we suggest experimenting with your little sister's poster paints. Hint: yellow and blue make green.

To find the product of two binomial (two-term) radical expressions, we use the distributive property. Remember him?

Sample Problem

Multiply .

The first thing we do is simplify each radical term, if possible. We can replace with 2, and now the problem is:

Now it's distribution time. We're gonna multiply the first term in the first set of parentheses by both terms in the second set, then multiply the -2x by both terms in the second set. It's all coming back to you now, right? Don't give us that look, everyone loves Celine.

Anyway, we start by multiplying the first terms:

That gives us:

Then we multiply that first term by the second term in the second set of parentheses:

...which gives us:

Now we distribute the -2x to both terms in the second set, starting with the square root of 3:

That gets us:

Finally, we multiply the last terms:

...which gives us our very last term:

Add together all of our cute little products to get our final answer:

Since the radicals have different radicands—one might even say "radically different radicands," which we will—this is as simplified as the answer gets. Rad.

Example 1

Multiply .


Example 2

Multiply .


Example 3

Multiply .


Exercise 1

Multiply, remembering to simplify: .


Exercise 2

Multiply, remembering to simplify: .


Exercise 3

Multiply, remembering to simplify: .


Exercise 4

Multiply, remembering to simplify: .


Exercise 5

Multiply, remembering to simplify: .


Exercise 6

Multiply, remembering to simplify: .


Exercise 7

Multiply, remembering to simplify: .


Exercise 8

Multiply, remembering to simplify: .


Exercise 9

Multiply, remembering to simplify: .


Exercise 10

Multiply, remembering to simplify: .


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