Fractional Exponents - Discovery Education

SD School Pre-Algebra
Program 3: Roots and Rational Numbers
QuikNotes
A cube root like a square root, except the number is
multiplied 3 times to get another whole number. The
cube root of 729 equals 9, because 9 x 9 x 9 = 729.
Student Notes
The 3 in the arm of the radical signals to us that we’re
looking for the cube root.
3
729 = 9
Whenever we have a cube root, or some other odd root,
it’s possible for us to end up with a negative number
under the radical. This is because whenever we multiply
an odd number of negative integers together, our
answer will be negative.
3
−8 =−2 because (−2) × (−2) × (−2) =−8
If there is a negative number under the radical and
we’re looking for an even root, like the square root of
negative 4, it is undefined. The reason we say it’s
undefined is because we can’t find one real number
multiplied by itself to equal negative 16.
This is different than when we have a negative sign in
front of our radical. Remember that this means we’re
looking for the negative of the square root of a number,
which we can do for any number. So, here the negative
square root of 16 is -4.
− 16 =−4
−16 is undefined
Raising a number to the fourth power means
multiplying that number four times. So, the fourth root
of a number means finding the number that when
multiplied 4 times gives us our original number.
For example:
4
16 = 2 because 2 × 2 ×2 × 2 = 16
When we’ve got a number with an exponent that’s a
simple fraction, like 3 to the one-half power, it just
means the square root of 3. We look at what number is
in the denominator and take that root of our base
number.
1
3 2
= 3

SD School Pre-Algebra
Program 3: Roots and Rational Numbers
QuikNotes
If the same kind of exponent is negative, take the
reciprocal of the base number, and then make the
exponent positive.
a
1
−
n
=
a
1
1
n
=
n
1
a
Student Notes
1
− 1 1 1
3
8 = = =
1 3
8 2
3
8
When given a fractional exponent where the numerator
is not a 1:
1. Raise the base number to the power of the
numerator.
2. Take the root as signaled by the denominator.
3
5
4 = 4 3
5 4 3 5 5
= 64 = 2 2
Irrational numbers are decimals that go on forever and
don’t repeat. An example would be the number pi (ð).
A repeating decimal is a rational number. An example
is the fraction, 1/3. It is equal to 0.3333333…
The repetition of 3 goes on forever.
A terminating decimal is also a rational number.
Rational, irrational, and repeating decimals are all real
numbers.
An inequality is a mathematical expression that is
unbalanced. Statements like 5 is greater than 3, or 4 is
less then 7 are considered inequalities because the two
sides don’t balance.
5 > 3 4 < 7

SD School Pre-Algebra
Program 3: Roots and Rational Numbers
QuikCheck
True or False
3 3
1. −8 =− 8 __________
2. − 16 = −16 __________
3. An exponent that’s a simple fraction means the number in the numerator is the root of our
base number. _____________
4. If a fractional exponent is negative, take the reciprocal of the base number and then make
the exponent positive. ________
5. Irrational numbers are decimals that go on forever and don’t repeat. ________
6. An inequality is a mathematical expression that when both sides are multiplied by zero,
yields zero on one side and an irrational number on the other. __________
Problem Solving
3
7. 216
=
1
10. 81 2
=
4
2
13. 9
3
8. −64
=
11. 36 −1 2
=
0
14. 7
3
4
9. 81
=
2
12. 8 3
=
2
3
15. −27

SD School Pre-Algebra
Program 3: Roots and Rational Numbers
QuikCheck Answer Key
True or False
3 3
1. −8 =− 8 True
2. − 16 = −16 False − 16 =−4 ; −16 is undefined.
3. An exponent that’s a simple fraction means the number in the numerator is the root of our
base number. False An exponent that’s a simple fraction means the number in the
denominator is the root of our base number.
4. If a fractional exponent is negative, take the reciprocal of the base number and then make
the exponent positive. True
5. Irrational numbers are decimals that go on forever and don’t repeat. True
6. An inequality is a mathematical expression that when both sides are multiplied by zero,
yields zero on one side and an irrational number on the other. False An inequality is a
mathematical expression that is unbalanced.
Problem Solving
3
7. 216
3
=
216 = 6
6 × 6× 6 = 216
3
8. −64 =
3
−64 =−4
−4×−4 ×−4 =−64
1
10. 81 2
=
1
81 2
= 81= 9
11. 36 −1 2
=
36 −1 2
= 1 36 = 1 6
4
2
13. 9
4
9 2
= ( 9) 4 = 3 4 = 81
0
3
14. 7
Any number to the 0 power = 1
0
7
3 3
= ( 7) 0 =1
4
9. 81 =
2
12. 8 3
=
2
3
15. −27
4
81 = 3
2
8
3 3
= ( 8) 2 = 2 2 = 4
− 27
2
3
= (
3
− 27)
2
= ( −3)
2
= 9
3× 3× 3× 3= 81

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