Multiplying

126

3155

A property of mathematics says that square roots can be

distributed over multiplication.

That means a radical such as

135

can be written as

135

Or a radical expression such as

36can be written as

218

which simplifies to6

Try these:

100520

1682

10

4

The property is often written:

abbaor

ab ba

A related property says that square roots are also

distributive over division:

ba

ba

ba

ba

or

2

502

50

Simplify:

25 5

3

273

27 9 3

Try one more:

75155 ?

Because 75 is not a perfect square . . .

75155

What makes this one different?

?

. . . we don’t know what its square root is.

We need to do something else to simplify it.

75

One possibility is to get an approximation

from a calculator

66.8

75

But another way to simplify it is to use what we just learned

about multiplying square roots.

325 32535

75

What just happened?What allowed us to change

to

35 ?

One of our numbers

25was the square root of

a perfect square.

Since we know the square root of perfect squares,

325we can write

as35

A radical term with all perfect square factors extracted from

the radical is said to be inSimplest Radical Form.

Look at this example:

Is it in simplest radical form?Have all perfect square factors been removed?

96 244 242

No—what is a perfect square factor of 24?

96 244 242

Did you say 4?We need to remove a factor of 4, so we will have:

96 242

You’re right!

642642 622 64

Just like when we simplify fractions,

when we simplify radicals, we are not through until all possible factors

have been removed.

And just like when we simplify fractions,

it saves a step or two if we remove the greatest possible

factor first.

Try this one:

31648 31634

Try this one:

295185295253 103

235

Try this one:

3

150 3

150 50

225 25