Download Radicals and Exponents

C H A P T E R
5
!
Radicals and
Exponents
Radicals
In order to simplify some expressions and solve some equations, you will need to find the square or cube root of
a number or variable. The radical symbol, !", signifies the root of a value. The square root, or second root, of x
2
is equal to !x", or !"x. If there is no root number given, it is assumed that the radical symbol represents the square
root of the number. The number under the radical symbol is called the radicand.
!
Adding and Subtracting Radicals
Two radicals can be added or subtracted if they have the same radicand. To add two radicals with the same radicand, add the coefficients of the radicals and keep the radicand the same.
2!2" + 3!2" = 5!2"
37
– RADICALS AND EXPONENTS –
To subtract two radicals with the same radicand, subtract the coefficient of the second radical from the
coefficient of the first radical and keep the radicand the same.
6!5" – 4!5" = 2!5"
The expressions !3" + !2" and !3" – !2" cannot be simplified any further, since these radicals have
different radicands.
!
Multiplying Radicals
Two radicals can be multiplied whether or not they have the same radicand. To multiply two radicals, multiply
the coefficients of the radicals and multiply the radicands.
(4!6")(3!7") = 12!42
", since (4)(3) = 12 and (!6")(!7") = !42
".
If two radicals of the same root with the same radicand are multiplied, the product is equal to the value of
the radicand alone.
(!6")(!6") = 6. Both radicals represent the same root, the square root, and both radicals have the same radicand, 6, so the product of !6" and !6" is 6.
!
Dividing Radicals
Two radicals can be divided whether or not they have the same radicand. To divide two radicals, divide the coefficients of the radicals and divide the radicands.
(10!15
")
!
(2!3")
10
!15
"
= 5!5", since !2! = 5 and !
= !5".
!3"
!3"
Any radical divided by itself is equal to 1; !
= 1.
!3"
!
Simplifying a Single Radical
To simplify a radical such as !64
", find the square root of 64. Look for a number that, when multiplied by itself,
equals 64. Since (8)(8) = 64, the square root of 64 is 8;!64
" = 8; !64
" is equal only to 8, not –8. The equation
x2 = 64 has two solutions, since both 8 and –8 square to 64, but the square root of a positive number must be
a positive number.
However, most radicals cannot be simplified so easily. Many whole numbers and fractions do not have roots
that are also whole numbers or fractions. You can simplify the original radical, but you will still have a radical in
your answer.
38
– RADICALS AND EXPONENTS –
To simplify a single radical, such as !32
", find two factors of the radicand, one of which is a perfect
square; !32
" = (!16
")(!2"); !16
" is a perfect square; the positive square root of 16 is 4. Therefore,!32
"=
(!16
")(!2") = 4!2".
!
Rationalizing Denominators of Fractions
An expression is not in simplest form if there is a radical in the denominator of a fraction. For example, the frac4
tion !
!3" is not in simplest form. Multiply the top and bottom of the fraction by the radical in the denominator.
4
!3"
!3"
!
!
Multiply !
!3" by !3" . Since !3" = 1, this will not change the value of the fraction. Since any radical multiplied by
4
4!3"
itself is equal to the radicand, (!3")(!3") = 3; (4)(!3") = 4!3", so the fraction !
in simplest form is !
3 .
!3"
!
Solving Equations with Radicals
Use the properties of adding, subtracting, multiplying, dividing, and simplifying radicals to help you solve equations with radicals. To remove a radical symbol from one side of an equation, you can raise both sides of the
equation to a power. Remove a square root symbol from an equation by squaring both sides of the equation.
Remove a cube root symbol from an equation by cubing both sides of the equation.
If !x" = 6, what is the value of x?
To remove the radical symbol from the left side of the equation, square both sides of the equation. In other
words, raise both sides of the equation to the power that is equal to the root of the radical. To remove a square
root, or second root, raise both sides of the equation to the second power. To remove a cube root, or third root,
raise both sides of the equation to the third power.
3
3
!x" = 6, (!x")2 = (6)2, x = 36; !x" = 3, (!x")3 = (3)3, x = 27
!
Exponents
When a value, or base, is raised to a power, that power is the exponent of the base. The exponent of the term 42
is two, and the base of the term is 4. The exponent is equal to the number of times a base is multiplied by itself;
42 = (4)(4); 26 = (2)(2)(2)(2)(2)(2).
Any value with an exponent of 0 is equal to 1; 10 = 1, 100 = 1, x0 = 1.
Any value with an exponent of 1 is equal to itself; 11 = 1, 101 = 10, x1 = x.
39
– RADICALS AND EXPONENTS –
!
Fractional Exponents
An exponent can also be a fraction. The numerator of the fraction is the power to which the base is being raised.
The denominator of the fraction is the root of the base that must be taken. For example, the square root of a num1
ber can be represented as x !2!, which means that x must be raised to the first power (x1 = x) and then the second,
1
or square, root must be taken; x !2! = !x".
3
4!2! = (!4")3 = 23 = 8
It does not matter if you find the root (represented by the denominator) first, and then raise the result to
the power (represented by the numerator), or if you find the power first and then take the root.
3
"3 = !64
"=8
4!2! = !(4)
!
Negative Exponents
A base raised to a negative exponent is equal to the reciprocal of the base raised to the positive value of that exponent.
1
!
3–3 = !
(33)
1
!
x–2 = !
(x2)
!
Multiplying and Dividing Terms with Exponents
To multiply two terms with common bases, multiply the coefficients of the bases and add the exponents of
the bases.
(3x2)(7x4) = 21x6
4
(2x–5)(2x3) = 4x–2, or !x!2
(xc)(xd) = xc + d
To divide two terms with common bases, divide the coefficients of the bases and subtract the exponents of
the bases.
(27x5)
!!
(9x)
= 3x4
(2x3)
!!
(8x4)
x–1
(xc)
!!
(xd)
1
!
= !4!, or !
(4x)
= xc – d
40
– RADICALS AND EXPONENTS –
!
Raising a Term with an Exponent to Another Exponent
When a term with an exponent is raised to another exponent, keep the base of the term and multiply the exponents.
(x3)3 = x9
(xc)d = xcd
If the term that is being raised to an exponent has a coefficient, be sure to raise the coefficient to the exponent
as well.
(3x2)3 = 27x6
(cx3)4 = c4x12
!
Practice
3
1.
2)
!(32x
"
4.
=
a. 4!2x
"
b. 4x!2"
c. 4x!8"
d. 16x
e. 16x!2"
a.
3.
c.
!3"
!
3
y!3"
!
3
d. y
e. y!3"
5.
2b)
2)
(!(a
")(!(ab
")
!!
!ab
"
a.
!ab
"
!
ab
"
b. !ab
c. ab
"
d. ab!ab
=
e. a2b2
a. 2
b. 4
6.
c. !g"
d.
=
b. !3"
3) =
2. a3!(a
"
4
a. a !a"
b. a5
c. a5!a"
d. a6
e. a9
!g"
!
4 !4g
"
(27y3)
!"
!
2)
!(27y
"
m
!
#!$
n
3
a.
2!g"
!
g
b.
e. 2!g"
c.
d.
e.
41
–2
5
m3
!!
n5
n5
!!3
m
m6
!1!0
n
n7
!!5
m
n10
!!
m6
=
=
– RADICALS AND EXPONENTS –
(ab)3
2
7. (!b!)4 =
a. a7
b. a12
c. a7b6
d. a12b8
e. a12b11
4
11. If a!3! = 6, then a!3! =
a. !3"
b. !6"
c. 3!6"
d. 6!6"
e. 36
1
!!
1
12. If (!p")4 = q–2, and q = – !3!, what is one possible
value of p?
8. ((4g2)3(g4))2 =
a. 8g3
b. 8g4
c. 8g5
d. 8g10
e. 8g12
1
a. – !3!
1
b. !9!
c.
1
!!
3
d. 3
9.
!9pr
"
!
– !3!
(pr) 2
e. 9
=
"
a. !3pr
1
13. What is the value of (a!b")–ab if a = !3! and b = 9?
3
b. !p!r
a.
c. 3!pr
"
b.
d. 3pr
c. 1
e. 3p2r2
x
10.
d.
d. 3
y
(!y!)2(!x!)–2
!
xy
1
a. !x!y
x3
b. !y!5
x3
c. !y!3
1
!!
9
1
!!
3
e. 9
=
14. What is the value of ((xy)y)x if x = 2 and y = –x?
a. –4
1
!
b. !
256
x3y3
c.
e. x5y5
1
!!
16
d. 4
e. 16
42
– RADICALS AND EXPONENTS –
!3"
18. What is the value of (x –y)(2x y)(3y x) if x = 2 and
y = –2?
a. 6
b. 8
c. 12
d. 24
e. 384
15. If g!108
"=!
g , what is a value of g?
a.
b.
c.
1
!!
36
1
!!
6
!6"
!
6
d. !6"
e. 6
"
!n
+5
16. If (c!d")2 = 48 and c = 2, what is the value of d?
a. 2!3"
b. 2!6"
c. 6
d. 4!6"
e. 12
17. If
20. If a is positive, and a2 = b = 4, what is the value of
b!b" a
(!
a4 ) ?
( )
a.
b.
!n
"
!
-!1!
n 2
n
!!
5
5
!
1
!!
n4
n
19. If n = 20, what is the value of !
(!2! !5")?
!n
"
a. 5
5!5"
b. !
2
c. 10
d. 5!5"
e. 25
m = 5 what is the value of m in terms of n?
a.
b.
1
!4!
c.
c. 5n
d. !n!
5
d.
e. 5!n
"
e.
43
1
!!
1,024
1
!!
32
1
!!
16
1
!!
4
1
!!
2