Exponents Summary

Zero Exponents

a0 = 1

Negative Exponents

a-n = 1 / an

Examples:

A) -5-2 = 1 / -52

b) 5a3b-2 =

5a3(1/b2) = 5a3/b2

What do I do if there is a negative exponent in the denominator (bottom)?

Example:

1 / x-5 = 1 / (1 / x5)

Next, we know that when we divide by a fraction, it is the same thing as multiplying by the reciprocal. So flip the bottom and multiply it by the top:

1 × x5 = x5

Multiplying Powers with the Same Base

am× an = am+n When youMULTIPLY the same bases with different exponents, just ADD the exponents

Hey – what if there are coefficients? Then multiply the coefficients.

Example:

3a2× 4a6 = 12a8

When you are multiplying and there are negative exponents, add or subtract the exponents before you make any fractions. You may not have to make a fraction. Here are 2 examples with each type:

4z5× 9z-3 Do not put z-3 into the denominator. Add or subtract the exponents first:

36z2

jNext example:

4z5× 9z-12 Again, add or subtract the exponents first (and multiply the coefficients)

36z-7Next, we form the fraction, but the 36 stays in the numerator (top), so we get:

36 / z7

Raising a Power to a Power

(am)n = amn Example: (a3)2 = a3x2 = a6

What if there is a COEFFICIENT inside the parenthesis?

That coefficient must also be raised to the power of the exponent outside the coefficient:

(ab)n = anbn (3×2)3 = 3323 = 27 x 8 = 216 This is also equal to:

63 = 216

Big Example:

Simplify: (n1/2)10(4mn-2/3)3

(n1/2)1043m3(n-2/3)3First, raise each factor of 4mn-2/3 to the 3rd power

n543m3n-2Multiply the exponents of a power raised to a power

Notice there is an n5 and an n-2. Add or Subtract exponents of

43m3n3Powers with the same base.

64m3n3Simplify.

Dividing Exponents

am / an = am-nTo divide powers with the same base, subtract the exponents

What if I have to raise a fraction, also known as a quotient, to a power?

You raise both the numerator and denominator to the power and then simplify.

Rule: (a/b)n = an / bn

Example:

(3/5)3 = 33/53 = 27/125

Big example:

Simplify: (2x6 / y4)-3Since the -3 exponent indicates that we will make this 1 over the whole

Expression, we can actually convert this to the reciprocal:

____1______

(2x6 / y4)3=(y4/2x6)3

(y4)3 / (2x6)3Raise the numerator and denominator to the 3rd power

y12 / 8x18Simplify

SCIENTIFIC NOTATION

Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers:

Example: 700

Why is 700 written as 7 × 102 in Scientific Notation ?

700 = 7 × 100

and 100 = 102 (see powers of 10)

so 700 = 7 × 102

Both 700 and 7 × 102 have the same value, just shown in different ways.

Example: 4,900,000,000

1,000,000,000 = 109 ,

so 4,900,000,000 = 4.9 × 109 in Scientific Notation

So the number is written in two parts:

Just the digits (with the decimal point placed after the first digit), followed by

× 10 to a power that puts the decimal point where it should be

(i.e. it shows how many places to move the decimal point).

In this example, 5326.6 is written as 5.3266 × 103,

because 5326.6 = 5.3266 × 1000 = 5.3266 × 103

To figure out the power of 10, think "how many places do I move the decimal point?"

When the number is 10 or greater, the decimal point has to move to the left, and the power of 10 is positive.

When the number is smaller than 1, the decimal point has to move to the right, so the power of 10 is negative.

Example: 0.0055 is written 5.5 × 10-3

Because 0.0055 = 5.5 × 0.001 = 5.5 × 10-3

Example: What is 1.35 × 104 ?

You can calculate it as: 1.35 x (10 × 10 × 10 × 10) = 1.35 x 10,000 = 13,500

But it is easier to think "move the decimal point 4 places to the right" like this:

1.35 / / 13.5 / / 135. / / 1350. / / 13500.

Example: What is 7.1 × 10-3 ?

Well, it is really 7.1 x (1/10 × 1/10 × 1/10) = 7.1 × 0.001 = 0.0071

But it is easier to think "move the decimal point 3 places to the left" like this:

7.1 / / 0.71 / / 0.071 / / 0.0071

MULIPLYING and DIVIDING with SCIENTIFIC NOTATION

What is( 1.13× 10-7) × (3.34 × 1022) ?

This might seem like a crazy problem, but actually we can use commutative and associative properties of multiplication to simplify this.

This expression can be rewritten as:

1.13 × 3.34 × 10-7 × 1022

3.7742 × 10-7+22

3.7742 × 1015

Division would just be dividing the real numbers and subtracting the exponents.

Laws of Exponents

Here are the Laws (explanations follow):

Law / Example
x1 = x / 61 = 6
x0 = 1 / 70 = 1
x-1 = 1/x / 4-1 = 1/4
xmxn = xm+n / x2x3 = x2+3 = x5
xm/xn = xm-n / x6/x2 = x6-2 = x4
(xm)n = xmn / (x2)3 = x2×3 = x6
(xy)n = xnyn / (xy)3 = x3y3
(x/y)n = xn/yn / (x/y)2 = x2 / y2
x-n = 1/xn / x-3 = 1/x3
And the law about Fractional Exponents: