Algebra 2 - Unit 8 – Fractional Exponents – NEED TO KNOW…..
A. Simplifying Radicals –
1. Perfect Radicands – The product of a number multiplied by itself however many times the given index is
called perfect. Examples:
4  2 (4 is a perfect square & 2 is the square root),
3
8  2 (8 is a perfect cube & 2 is the cube root),
5
32  2 (32 is perfect & 2 is the 5th root).
2. Index – The number that indicates how many times the root must be multiplied by itself to be perfect. It is
on the outside top left of the radical. If no index is indicated then 2 is assumed to be the index.
Examples:
4  2 (since no index then 2 is the index so 2 multiplied by itself is 4),
8  2 (3 is the index so 2 multiplied by itself 3 times is 8),
5
32  2 (5 is the index so 2 multiplied by itself 5 times is 32).
3
3. Root – This number multiplied by itself however many times indicated by the index is called the root of the
perfect radicand.
4. Radicand – Number under the radical symbol.
5. Simplifying Radicals – To simplify a radical try to rewrite the radicand in exponential form with a base and
an exponent that matches the index, if any exist, then take the root of the perfect radicand and move the root
out from under the radical and leave under the radical factors that are not perfect .
4  22  2 ; 3 8  3 23  2 ; 4 48  4 24  3  2 4 3
6. Simplifying Radicals with Variables – Remember radicals with variables with exponents that match
the index are perfect. If you have a radical with a variable with an exponent, the
root of the radical is the variable with the exponent divided evenly by the index. If you have a
radical with a variable with an exponent, find the factors of the variable so that you have a variable
product that contains the exponent that matches the index. To do this just divide the exponent by
the index and then you have 2 factors one with an exponent matching the index and one with an
exponent that does not match the index. When simplified the radical is the root of the variable with
the exponent with the matching index multiplied by the radical with the variable with the exponent
that does not match the index.
Examples:
Examples:
x 6  ( x3 )2  x3 ;
4
x 27  4 ( x 6 ) 4  x 3  x 6 4 x 3
B. Rules to Live by (Used for all indexes not just the index of 2) -
1. Multiplication -
a  b  a b
a
a

b
b
3. You can only Add or Subtract LIKE Radicals (under the radical has to be identical!).
4. NO Radicals can be left in the Denominator!
5. To Rationalize the Denominator –
If you have one term in the denominator –
Multiply the numerator and the denominator by the multiple of 1 that
2. Division -
a
represents
denominator index-exponent
and then simplify.
denominator index-exponent
Where “a” represents the index.
If you have a binomial in the denominator –
Multiply the numerator and the denominator by the multiple of 1 that
conjugate
represents
and then simplify.
conjugate
6. HINT – Simplify before you start rationalizing(it will save you time later).
a
Algebra 2 - Unit 8 – Fractional Exponents – NEED TO KNOW…..
C. Fractional Exponents – An alternate notation for the nth Root of a number;
1. Definition 1
If a is a real number and n is a positive number then a n  n a
(Translated - a to the 1/n equals the nth root of a.)
If a is a real number and m & n are positive numbers then
m
1
m
1
a n  (a m ) n  n a m , or , a n  (a n )m  ( n a )m
m
n
(Translated – a equals the nth root of am or equals (nth root of a) raised to the mth power.)
Hint: You need to look at the problem to see which form would be easier to use.
2
1
Example: 27 3  (272 ) 3  3 27 2  3 (33 )2  3 (32 )3  (32 )  9
BUT it is much easier to use the other definition to evaluate
2
1
27 3  (27 3 )2  ( 3 27)2  ( 3 33 )2  (32 )  9
2. Rules for Fractional Exponents are the SAME as for any other Exponents –
Rules & Definitions of Exponents: m & n are REAL Numbers
Multiplication - x m  x n  x m  n
xm
Division - n  x m  n
x
Exponents Raised to Exponents -  x m   x mn
n
Exponential Expressions Raised to Exponents -  x m y n   x m p y n p
p
0 as an Exponent -
x0  1
00 undefined
1
xm  m
x
Negative Exponents 1
 xm
xm
3. Examples –
Simplifying –
48 x 2 y 5  3 23  6  x 2  y 3  y 2  2 y 3 6 x 2 y 2
Adding/Subtracting –
3
2 3  5 12  27  2 3  5 22  3  32  31  2 3  10 3  3 3  5 3
Multiplying –
2 4
2  3  2  3 2  4 2  12  14  7 2



Dividing –
a(b2 )2
ab4
2 b 2 2a



8
4
2 2
2
5 3
5  3 4  5 ( 5  3)(4  5) 4 5  5  12  3 5
5 7





16  5
11
11
4 5 4 5 4 5
Algebra 2 - Unit 8 – Fractional Exponents – NEED TO KNOW…..
4. Solving Radical Equations –
a.) Property of Raising Each Side of Equation to a Power –
If 2 numbers are equal then the same powers of the numbers are equal.
Examples: 2  2, so, 23  23 ; 3  3, so,32  3x1 , then2  x  1& x  1 .
b.) If you are solving a radical equation by raising each side to a power then you must CHECK
each solution due to the possibility that Extraneous Solutions may exist.
c.) Hints to Solve – 1.)Get Radical by itself.
2.)Try to find the same base so you can do some simplifying before
you solve the problem.
3.)Once radical is by itself then raise each side of the equation to a
power equal to the index of the radical.
4.)Now use all your rules & definitions to solve the equation.
5.)CHECK your solutions in the original radical equation to make
Sure there are no extraneous solutions.
Examples:
1.) 2 x2  16, 2 x2  24 , so, x  2  4 & x  2. CHECK 22 2  16,16  16.
2.)
4
x  8  3, ( 4 x  8) 4  34 , x  8  81, so, x  89
CHECK
4
3.)
4
89  8  3, 4 81  3, 4 34  3,3  3 .
y 2  y  4  2, ( 4 y 2  y  4) 4  24 , y 2  y  4  16, y 2  y  12  0, ( y  4)( y  3)  0
y  4, or , y  3
CHECK
1
4
(4) 2  4  4  2, 4 16  2, 2  2 ;
4
(3) 2  3  4  2, 4 16  2, 2  2 so both OK.
1
1
1
1
4.) x 3  2; 3  2;1  2 3 x ;  3 x ;( )3  ( 3 x )3 ;  x
2
2
8
x
1
1
CHECK ( ) 3  2; 2  2
8