Tutorial 4: Radical

The thought behind this is that we are looking for the square root
of 81. This means that we are looking for a number that when we square
it we get 81.

What do you think it is?

**Let's find out if you are right:**

Since 9 squared is 81, **9 is the square root of 81.**

Note that we are only interested in the principal root and since 81
is positive and there is not a sign in front of the radical, our answer
is positive 9. If there had been a negative in front of the radical
our answer would have been -9.

The thought behind this is that we are looking for the cube root of
-27. This means that we are looking for a number that when we cube it we
get -27.

What do you think it is?

**Let's find out if you are right:**

Since -3 cubed is -27, **-3 is the cube root of -27.**

Since it didn't say that *x *is positive,
we have to assume that it can be either positive or negative. And
since the root number and exponent are equal we can use the rule.

Since the root number and the exponent inside are equal and are the
even number 2, we need to put an absolute value around* x* for our answer.

The reason for the absolute value is that we do not know if *x* is positive or negative. So if we put *x *as
our answer and it was negative, it would not be a true statement.

Now we are looking for the square root of -4, which means we are looking
for a number that when we square it we get -4.

What do you think it is?

**Let's find out if you are right:**

Since there is no such real number that when we square it we get -4,
then **the answer is not a real number.**

Note that both radicals have an index number of 2, so we were able
to put their product together under one radical keeping the 2 as its index
number.

Since we cannot take the square root of 14 and 14 does not have any
factors we can take the square root of, this is as simplified as it gets.

In this example, we are using the product rule of radicals in reverse
to help us simplify the cube root of .
When you simplify a radical, you want to take out as much as possible.
The factor of that we can take the square root of is . We can write as and then use the product rule of radicals to separate the two numbers.
We can take the square root of ,
which is , but
we will have to leave the rest of it under the square root.

***The cube root of -1 is -1 and the cube root
of 8 is 2**

***Simplify the fraction**

***The square root of 9 a squared is 3|a|**

Both radicals are as simplified as it gets.

Note how both radicals are the square root of *x*.
These two radicals are like radicals.

The 20 in the first radical has a factor that we can take the square
root of.

Can you think of what that factor is?

**Let's see what we get when we simplify the first radical:**

***Square root of 4 is 2**

Note how both radicals are the square root of 5. These two radicals
are like radicals.

Since we have a square root in the denominator, ** we need to
multiply by the square root of an expression that will give us a perfect
square under the radical in the denominator.**

Square roots are nice to work with in this type of problem because if the radicand is not a perfect square to begin with, we just have to multiply it by itself and then we have a perfect square.

So in this case we can accomplish this by **multiplying top and bottom
by the square root of 7:**

***Den. now has a perfect square under sq. root**

**AND**

In general the conjugate of* a* + *b *is *a* - *b* and vice versa.

**So what would the conjugate of our denominator be?**

It looks like the conjugate is .

***Use distributive prop. to multiply the numerators**

***In general, product of conjugates is **

**AND**

***Square root of 2 squared is 2 and the square
root of 3 squared is 3**

***Divide BOTH terms of num. by -1**

Last revised on Dec. 6, 2009 by Kim Seward.

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