Exponents and roots
a to the third power b to the third power is the same as
(ab) to the third power
10 squared
100
10 cubed
1000
10 to the third power
1000
12 squared
144
2 to the 4th power
16
4 squared
16
7 squared
49
2 to the 6th power
64
4 cubed
64
8 squared
64
2 cubed
8
2 to the third power
8
3 squared
9
30 squared
900
When you apply an exponent to an entire fraction
Apply the exponent separately to the top and to the bottom of the fraction
When you apply an exponent to a product
Apply the exponent to each factor in the product
To simplify square roots
Factor out squares; pull out perfect squares from under the radical sign; if you don't spot the perfect squares, you can always do a factor tree and work your way down to the primes. Any pair of primes under a radical sign becomes a single copy of that prime outside the radical.
4 squared
Is 16; or 2 to the 4th power
4 to the first power
Is 4; or 2 squared
4 to the third power
Is 64; or 2 to the 6th power
A square root of a positive number raised to a power...
Is equivalent to an exponent of 1/2; so you'd rewrite the square root as an exponent of 1/2 and then multiply the exponents.
A negative power (exponent)
Is one over a positive power; in other words a to the negative 2 is equal to the reciprocal of a to the 2nd power. a to the negative 2 = 1 over a squared
Divide terms with the same base
Subtract the exponents
If you multiply two bases that have exponents (and the bases are the same)
You add the exponents
When you multiply or divide square roots....
You can combine everything under one radical sign and then simplify; the square root of a times the square root of b is the same as the square root of ab; the square root of a divided by the square root of b is the same as the square root of a over b
As long as you're only multiplying and dividing...
You can simplify the expressions; DO NOT simplify if you're adding or subtracting roots, you can only do that when multiplying or dividing.
When you square a square root
You get the original number (the number underneath the radical)
1 cubed
1
1 squared
1
Multiply terms with the same base
Add exponents (you have to make sure the bases are the same)
Negative number raised to an odd power
Always negative
Negative number raised to an even power
Always positive
Since an even exponent always gives a positive result
An even exponent can hide the sign of the base; always be careful when dealing with even exponents in equations. Look for more than one solution. 4 squared is 16, but so is -4 squared. 4 squared and negative 4 squared are both equal to 16.
When you divide square roots...
you can put everything under the same radical sign; this works the other way too; you can separate numbers that are all underneath a radical sign. For example, the square root of 27 divided by the square root of 3 is the same as the square root of (27 over 3); or the square root of (27 over 3) is the same as the the square root of 27 divided by the square root of 3.
11 squared
121
5 cubed
125
5 to the third power
125
2 to the fifth power
32
6 squared
36
2 squared
4
If you have two factors with the same exponent...
Might want to regroup the factors as a product; 2 to the 4th power x 3 to the 4th power can be rewritten as (2 x 3) to the 4th power. In other words, a to the 3rd power x b to the 3rd power, is the same as (a x b) to the 3rd power
If you just have one base raised to two successive powers
Multiply the exponents (multiply the powers); the exponents can be positive, negative, one of each. Either way, multiply the exponents.
When you raise something that already has en exponent to another power
Multiply the two exponents together
When you multiply separate square roots...
You can just put everything under the same radical sign; this works the other way as well; you can separate numbers that are being multiplied underneath a radical sign. For example the square root of 8 times the square root of 2 is the same as the square root of (8 times 2); or the square root of (8 times 2) is the same as the square root of 8 times the square root of 2.
3 to the 4th power
81
9 squared
81
If you take the square root of 1 or 0
You end up with the number you started out with; because the square root of 1 is 1, and the square root of 0 is 0
When you square root a square
You get the positive value of the original number
13 squared
169
14 squared
196
Anything to the power of Zero is equal to...
1: the only exception to this is zero... zero to the zero power is undefined. Every other number, no matter what form, raised to the zero power is equal to 1
15 squared
225
5 squared
25
3 cubed
27
3 to the third power
27
2 squared equals
4
20 squared
400
If you raise a number to a fractional power...
Apply two exponents - the numerator as a power and the denominator as a fractional root, in whatever order seems easiest to you. For example, 125 to the 2/3 power would become the (cubed root of 125) squared. Or said differently, 125 squared and then the cubed root of that
When adding or subtracting terms with the same base
Pull out a common factor; remember when the bases are the same and you have exponents, but you're adding the bases, do not make the mistake of adding the exponents. That is incorrect. You only add the exponents when you're multiplying bases that are the same. When you're adding or subtracting terms with the same base, pull out a common factor.
Add or subtract under the root symbol
Pull out a square factor from the sum or difference. Remember that you cannot break the root into ___ plus ___;You can break up products, not sums. If the numbers you're adding underneath the radical are small, you can crunch the numbers. If they're big, factor Out a square factor.
To get rid of negative exponents in a fraction
Rewrite the expression using positive exponents; if the negative exponent is in the numerator, move the term to the denominator and the exponent becomes positive; if the negative exponent is in the denominator, move the term to the numerator and the exponent becomes positive. DO NOT confuse the sign of the base with the sign of the exponent; a negative base still remains negative when you move it from numerator to denominator for example... only the sign of the exponent switches. If you move the entire denominator, make sure you leave a 1 behind. Same goes for the numerator, if you move the entire numerator to get rid of the negative exponent, leave a 1 behind.
If you have different bases that are numbers
Try breaking down the bases to prime factors; you might discover that you can break everything down to one base. For example: 2 squared x 4 cubed x 16 can be rewritten as 2 squared x 2 squared to the 3rd power x 2 to the 4th power. Then 2 squared to the third power (multiply the two powers) becomes 2 to the 6th power. Now all three numbers have the same base. You can now add all the exponents (since all the bases you're multiplying against each other are the same... when multiplying bases with exponents, if bases are the same, add the exponents)
When you take the square root of any number greater than 1...
Your answer will always be less than the original number
When you take the square root of a number between 0 and 1....
Your answer will be greater than the original number
ab to the 3rd power is the same as
a to the third power b to the third power
