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Exponents

An exponent says how many times a number (the base) is to be used in a product. In this expression,

aba^b

a is the “base” and b is the “exponent.” 

Example:

323^2

In this case, 3 is the base, and the exponent of 3 is 2, so 3 is used in the product 2 times. This means the same thing as .

3×33\times 3

Thus,

32=3×33^2 = 3 \times 3

which is equal to 9.

Another example:

2×32×532 \times 3^2 \times 5^3

This is equal to

2×3×3×5×5×52 \times 3 \times 3 \times 5 \times 5 \times 5

That is,

the exponent of 2 is 1, so 2 is in the product once;

the exponent of 3 is 2, so 3 is in the product twice;

the exponent of 5 is 3, so 5 is in the product 3 times.

Multiplying numbers with the same base

Example:

32×333^2 \times 3^3

In this case, the base is 3, and the first time its exponent is 2, and the second time its exponent is 3. This can be expressed as:

(3×3)×(3×3×3)=3×3×3×3×3(3 \times 3) \times (3 \times 3 \times 3) = 3 \times 3 \times 3 \times 3 \times 3

In other words, 3 is a factor in the product 5 times. So, the original expression

32×333^2 \times 3^3

is equal to

353^5

It’s easy to see that if you multiply two numbers with the same base, each having an exponent, then the result will be the base with the two exponents added together:

32×33=32+3=353^2 \times 3^3 = 3^{2+3} = 3^5

In general, if the base is a, and b and c are exponents, then:

ab×ac=ab+ca^b \times a^c = a^{b+c}

Dividing numbers with the same base

Example:

5453\frac {5^4}{5^3}

In this case, the base is 5, and its exponent in the numerator is 4, while its exponent in the denominator is 3. This can be expressed as:

5×5×5×55×5×5\frac{5 \times 5 \times 5 \times 5}{ 5 \times 5 \times 5}

Notice that the 3 instances of 5 in the denominator cancel out with 3 of the instances of 5 in the numerator, leaving just one 5 in the numerator. In other words:

5×5×5×55×5×5=51=5\frac{5 \times 5 \times 5 \times 5}{ 5 \times 5 \times 5} = 5^1 = 5

It’s obvious that this will always work for fractions of numbers having the same base, so we can make the following rule:

If the base is a, and b and c are exponents, then:

abac=ab−c\frac{a^b}{a^c} = a^{b-c}

Or, as in the example above: 

5453=54−3=51=5\frac{5^4}{5^3} = 5^{4-3} = 5^1 = 5

Negative exponents

What happens if we’re dividing numbers with the same base, and the exponent in the denominator is larger than the exponent in the numerator?

Example:

5254\frac {5^2}{5^4}

Then the result is 

5−25^{-2}

But if we didn’t subtract exponents, but instead just canceled out 5’s from the numerator and denominator we would get

5254=152\frac {5^2}{5^4} = \frac{1}{5^2}

So, we conclude that 

5−2=1525^{-2} = \frac{1}{5^2}

In general, if the base is a, and b is an exponent, then:

a−b=1aba^{-b} = \frac{1}{a^b}

Zero exponents

What happens if we’re dividing numbers with the same base, and the exponents in the numerator and denominator are equal?

Example:

5454\frac {5^4}{5^4}

Then the result is 

505^0

But if we didn’t subtract exponents, but instead just canceled out 5’s from the numerator and denominator we would get

5454=11=1\frac {5^4}{5^4} = \frac{1}{1} = 1

So, we conclude that if the base is a, and the exponent is 0, then:

a0=1a^0 = 1

In other words, any number to the 0 power equals 1.

(An exception to this rule is 0 to the 0 power. This is undefined.)

Fractional exponents

Suppose we break an exponent into parts like this:

Example:

41=412+12=412×4124^1 = 4^{\frac{1}{2} + \frac{1}{2}} = 4^{\frac{1}{2}}\times 4^{\frac{1}{2}}

But can we figure out what 

4124^{\frac{1}{2}}

is? We see from above that

4=41=412×4124 = 4^1 = 4^{\frac{1}{2}}\times 4^{\frac{1}{2}}

so

4124^{\frac{1}{2}}

must be the square root of 4 (which is 2).

In general, the rule is:

If the base is a, and b and c are integers, then:

abc=abca^{\frac{b}{c}} = \sqrt[c]{a^b}

Raising to powers

Example: 

(32)4(3^2)^4

This expands to

32×32×32×323^2 \times 3^2 \times 3^2 \times 3^2

and each of these expands to

3×33 \times 3

so, the final result is

3×3×3×3×3×3×3×33 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3

In other words, 3 appears in the product 8 times.

Therefore, the general rule is if the base is a, and b and c are integers, then:

(ab)c=ab×c(a^b)^c = a^{b \times c}

See also Logarithms