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