RADICALS
For every operation there is an opposite that can be used to cancel the operation or make it equal to
zero.
Example:
+
×
inverse
–
inverse
÷
x2
x
3
x
inverse
inverse
3
x
The opposite of exponents are called radicals. If the exponent is a square, or 2, the opposite would be a
square root; if the exponent is 3, the opposite would be a cubic root, and so on.
Examples:
22 = 4
23 = 8
inverse
inverse
35 = 243
inverse
3
5
4=2
8=2
243 = 3
Radicals can also be written in exponent notation; however, in this case the exponent would be a
fraction.
4 = 41 2
Examples:
3
8 = 81 3
5
243 = 2431 5
The expression 3 8 is called a radical expression, where 3 is called the index,
is the radical
sign, and 8 is called the radicand. The index of a radical expression must always be a positive integer
greater than 1. When no index is written it is assumed to be 2, or a square root.
Negative Radicals – The only restriction that exists for negative signs and radicals is that there cannot
be a negative sign under an even root since there is no real solution to this problem. However, a
negative sign can exist in front of a radical or under odd roots and still be able to obtain a real number.
Examples:
3
− 16
← Negative sign under even root, no real answer.
(4)(4) = 16 or
(− 4)(− 4) = 16
−8
← Negative sign under odd root, real answer.
(− 2)(− 2)(− 2) = −8
− 81
← Negative sign is outside the radical, the sign does not affect the
Calculations. It is only carried out to the final answer.
− 81 = −(9 )(9 ) = −81
Radicals Containing Large Numbers – When working with large numbers under a radical sign, it is
always easier to break down the number and work in smaller parts rather than trying to find an exact
root.
← Break down the number into smaller numbers.
525
Example:
21 ⋅
← Find the root of each of the smaller numbers.
25
← Write final answer with single digits first and radicals
at the end.
21 ⋅ 5
5 21
Sometimes it is not easy to find two small numbers that when multiplied give you the large numbers.
In these cases, you may break down the large number little by little.
← Follow the same procedure as described above. It
takes a little longer, but at the end the same result is
obtained.
525
Example:
105 ⋅ 5
105 ⋅
5
21 ⋅ 5 ⋅
21 ⋅
5
25
21 ⋅ 5
5 21
Multiplication and Division of Radicals
Radical multiplication is similar to polynomial multiplication, the same distributive property applies.
For instance, if you had to multiply 3(x + 2), you would take the 3 through the parentheses to get 3x +
6. With radicals it works the same way.
Examples:
(
)
3 2 3+ =
5
(
)
3 2 3 + 3
( 5)
→ Distribute the outside radical to each
term inside the parentheses.
= 2 9 + 15 = 2 ⋅ 3 + 15
= 6 + 15
(
3+ 5
)(
)
3 − 6 = 9 + 15 − 18 − 30
= 3 + 15 − 9 ⋅ 2 − 30
=
3 + 15 − 3 2 − 30
Rationalizing the Denominator
When dealing with fractions, a final answer cannot contain radicals in the denominator; therefore, it is
necessary to eliminate any radical from the denominator. The process of removing the radical from the
denominator is called rationalizing the denominator.
Examples:
Simplify the following expression:
2
3
To simplify the expression we need to eliminate the radical in the denominator by
rationalizing the denominator. It is necessary to remove the radical sign from 3 and
leave the 3 by itself. To do this, multiply
3 by another
3 and you'll get 3.
x = x , and when you multiply the exponents add together.
Therefore, x ⋅ x = x or x1 2 ⋅ x1 2 = x . However, don’t forget to multiply the
numerator as well to compensate the change.
12
Remember that
2
2
3 2 3
=
⋅
=
3
3
3 3
→ Multiply the denominator by 3 to eliminate the
radical.
2
3+ 6
→
When the denominator is a little more complex, multiply the
denominator by its conjugate and the radicals will disappear. The
conjugate is the same denominator but with a different sign in
between: in this case 3 − 6 .
2
2
3− 6
=
⋅
3+ 6 3+ 6 3− 6
=
=
Simplify:
1.
27
2.
72
3.
4
4.
5.
16
50
3
250
6.
1
5
7.
3
5
8.
( 2 + 3 )( 2 − 3 )
9.
5
2+ 3
10.
11.
6
5− 2
x− y
x+ y
12. 4 2 −
1
+ 32
8
(
2 3− 6
)
9−3 6 +3 6 − 3 6
6−2 6 6−2 6
=
9−6
3
RADICALS - EXERCISES
13. 6 + 4 2 − 2 + 3
3 + 27
14.
(3
15.
6− 3
)
2
RADICALS - ANSWERS TO EXERCISES
1.
27 = 3 ⋅ 9 = 3 3
2.
72 = 36 ⋅ 2 = 6 2
3.
4
50 = 25 ⋅ 2 = 5 2
4.
5.
16 = 2
3
250 = 3 125 ⋅ 2 = 53 2
6.
5
5
7.
15
5
8. 1
9. 10 − 5 3
10. 2 5 + 2 2
11.
x2 − 2x y + y
x2 − y
12.
31 2
4
13. 9 + 3 2
14. 4 3
15. 57 − 18 2