Download Radical Expressions

Radical Expressions
Square roots, cube roots and fourth roots
Recall how to evaluate the square root of a number:
√ 36 equals
_______
because ________ = 36
√ 81 equals
_______
because ________ = 81
√−9 is _________________ because any real number, when ______________________
must equal a positive number.
This works with variables, too. Assume all variables are positive.
√ x6
equals
√ 16 a2
_______
equals
_______
because ________ =
x6
because ____________ =
16 a
2
A few notes about square roots (and all even roots, as we'll see later):
•
The result of a square root (or fourth/sixth root) is always positive (called the principal root).
•
The radicand (inside) of a square root must be positive to produce a real-valued answer.
•
Odd roots (third/fifth, etc) don't have these limitations.
•
In general,
√n xn
= x , and more specifically,
√n xn
= |x| if n is even.
Evaluating/simplifying cube roots is similar to square roots:
√3 27 equals
√3 1 equals
√3 −8 equals
_______
_______
_______
because ________ = 27
because ________ = 1
because ________ = -8
*** The radicand and result of a cube root may be negative !!! ***
1
Working with variables is similar:
√3 x6
equals
√3 −64 a3
_______
equals
because _______________________ =
_______
x6
because ________________________________ =
3
−64 a
And, higher index roots work in much the same way:
√4 81 equals
_______
because _______________________ =
81
√4 −16 is ___________________ because ____________________________________________ .
√5 −32 x 5
equals
__________ because ________________________________ =
−32 x
5
Perfect Squares, Cubes and Fourths
We already examined this topic when we factored.
Perfect Squares: 1,
4,
9, _____, _____, _____, _____, _____, _____, _____, _____
Perfect Square Variables:
x2,
Perfect Cubes:
_______,
1,
Perfect "Fourths":
8,
1,
x4 ,
x6
(any variable where the exponent is ________ ).
________,
_______ , ________ , ...
_______ , ...
4
x ,
8
x ,
x3 ,
x6 ,
______ , _______
_______
2
Simplifying Radicals
If the radicand contains perfect powers only, the process is easy. Just take the nth root of each factor.
For even roots, assume variables are positive.
Ex1.
√ 49 a6 b 14
Ex2.
√3 125 x 12 y 3
If the radicand contains non-perfect powers, we simplify by rewriting the radicand as the product of the
largest perfect power factor times a non-perfect factor.
Ex1.
√ 48
Ex2.
√ 200 x 4
48 is not a perfect square. Find the largest perfect square factor of 48.
3
Ex3.
√ 25 a7 b3
Ex4.
√3 −81
Ex5.
√3 32 x 10 y 12
Ex6.
√4 32 x 10 y 12
4
Connection Between Radicals and Exponents
The fact that we divide exponents by the index of the root suggest a connection between radicals and
exponents. In particular, the following are true:
n
√a = a
1
n
and
n
√a
m
n
m
= ( √ a)
= a
m
n
So this means we can rewrite radicals in exponential form, and vice versa.
Ex1.
a)
Rewrite the following in exponential form:
√5
Ex2.
b)
5
√ x3
Evaluate the following. Use any and all rules of exponents
1/2
a)
16
c)
27 −1/ 3
b)
d)
2/3
8
4 1/ 2
9
()
5
Ex3. Simplify using basic rules of exponents.
a)
x ⋅ x 1/3
b)
x1/ 2
8
( )
y 3/4
Adding and Subtracting Radical Expressions
You can combine "like radicals" with addition and subtraction by combining coefficients.
Like Radicals means all terms have the same ________________ and _______________________
as well as the same non-radical variable parts.
Ex1.
Simplify:
19 √ y −12 √ y
Ex2.
Simplify:
3 x √4 7 + 5 x
√4 7
6
√ 32 + √12 + 7 √ 2
Ex3.
Simplify:
Ex4.
Simplify:
7 √3 81 − 4 √3 16 − 5 √3 2
Ex5.
Simplify:
4 x √ 2 x + 3 √ 2 x3
7
Multiplying and Dividing
Basic principal (both assume that
n
n
n
√ a ⋅ √ b = √a⋅b
√n a and √n b are real numbers):
and
√n a = =
√n b
√
n
a
b
General approach is to combine first, then simplify the result.
Ex1. Simplify:
√3 6 x2 y 3 ⋅ √3 9 x 3 y
Ex2. Simplify:
√18 x 5 y
√2 x y 6
Ex3.
(3 − 2 √8)(6 + 3 √ 2)
Simplify:
8
2
Ex4.
Simplify:
( √ x + 2 √ y)
Ex5.
Simplify:
( √ 3+ √ 5)( √ 3−√ 5)
Notice that the middle terms cancelled.
We say that (a + b) and (a - b) are ________________________ of each other. When we multiply
conjugates together, the middle terms always cancel out.
Note: we don't consider a radical expression to be simplified if there is a radical in the denominator.
In this situation, we need to ____________________________ the denominator.
Ex1. Simplify:
3
√2
9
Ex2.
Simplify:
21
√7 x
Ex3.
Simplify:
9
√ 4 a2
3
If the denominator has two terms and at least one is a square root, you may be able to rationalize by
multiplying numerator and denominator by the conjugate.
Ex4.
Simplify:
5
√3−√ 2
Ex5.
Simplify:
3+ √ 5
1−√ 5
10
Solving Radical Equations
An equation containing radicals can be solved by isolating the radical, and then raising both sides of
the equation to the same power. This is based on the following observation:
you take an n-th root the the n-th power, the radical "disappears".
n
( √n expr ) = expr . If
√ 5 x−2 = 4
Ex1.
Solve:
Ex2.
Solve:
√4 7 x −5 = 2
Ex3.
Solve:
10 − √3 2 x +4 = 13
Make sure you isolate the radical first!
11
Ex4.
Sometimes there will be no solutions. Solve:
√ 3 x +1 = −5
Ex5.
Or one of the solutions won't work:
x = 1 + √2 x+6
Solve
We call apparent solutions that don't work "extraneous roots". These are false solutions that pop-up
during the process of squaring both sides of an equation.
12