Enhanced Learning Center Quick Reference
Working with Exponents
Math
What does an exponent do?
x
3 is the same as x · x · x .
More generally, x n is the same as x · x · x · · · x (n times).
Negative Exponents: When we encounter a negative exponent, we can switch the side of the fraction it is on (numerator to denominator and vice-versa) and change the sign of the exponent.
So, x
− 4 = x
− 4
1
=
1 x 4 and
1 x
− 2
= x
2
1
= x 2 .
Roots: n-roots of a number (square roots have n = 2) can be re-written as fractional exponents.
This is often very helpful in solving equations and simplifying expressions.
√ x = x
1
2
√
4 x = x
1
4
√
3 x 2 = x
2
3
More generally,
√ n x m = x m n
Multiplying with Exponents: When multiplying together different powers of the same variable , sum the exponents.
x 3 · x 2 = x 3+2 = x 5
This is intuitive because x
3 · x
2
= ( x · x · x ) · ( x · x ) = x · x · x · x · x = x
5
.
This holds even if the exponents are fractions.
x
2 · x
1
2
= x
2+
1
2
= x
5
2 x
3 · x
− 4 · x
5
= x
3 − 4+5
= x
4
Dividing with Exponents: When dividing powers of the same variable, subtract the bottom exponent from the top exponent.
x
3 x 2
= x
3 − 2
= x
1
= x
1

Enhanced Learning Center Quick Reference Math
One could think of this as x
3 x 2
= x · x · x x · x
= x x
· x x
· x = 1 · 1 · x = x .
Also, x 3 x
− 4
= x
3 − ( − 4)
= x
3+4
= x
7 and x
3 x 5
= x
3 − 5
= x
− 2
=
1 x 2
Exponents with Exponents When a variable raised to a power is then raised to another power, simply multiply together the two exponents.
( x
3
)
2
= x
3 · 2
= x
6
This makes sense because
( x
3
)
2
= ( x
3
) · ( x
3
) = ( x · x · x ) · ( x · x · x ) = x · x · x · x · x · x = x
6
.
The root rules from earlier can also be thought of in this way.
√
3 x 6 = ( x
6
)
1
3
= x
6 ·
1
3
= x
6
3
= x
2
Contributed by Sarah Withem, E.L.C. Tutor
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