**Review of Exponents**

- Exponents are the mathematician’s shorthand.
- In general, the format for using exponents is:

(base)^{exponent}

where the exponent tells you how many of the**base**are being multiplied together. - Consider: 2 • 2 • 2 is the same as 2
^{3}, since there are**three 2’s**being multiplied together.

Likewise, 5 • 5 • 5 • 5 = 54, because there are**four 5’s**being multiplied together. - Exponents are also referred to as “powers”.

For example, 2^{3}can be read as “two cubed” or as “two raised to the third power”.

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**Exponents of Negative Values**

When we multiply** negative** numbers together, we must utilize parentheses to switch to exponent notation.

(-4)(-4)(-4)(-4)(-4)(-4) = (-4)^{6}

The examples below show that (-4)^{6} and -4^{6} are not the same.

The missing parentheses mean that **-4 ^{6}** will multiply

**six 4’s**together first (by order of operations), and then

**take the negative of that answer.**

**Example: **(-3)

^{2}= (-3) × (-3) = +9

The 2 is “attached” to the parentheses, so everything inside the parentheses is squared.

-3^{2} is not the same as (-3)^{2}

-3^{2} = -(3^{2}) * *The 2 is “attached” to the 3, but not to the negative sign.

*-3*

^{2}= -(3

^{2}) = -(3 × 3) = -(9) = -9

The expression -3

^{2}(with missing parentheses) means to multiply two 3’s together first (by order of operations), and then take the negative of that answer.

**Even powers of negative numbers** allow for the negative values to be arranged in pairs. This pairing guarantees that the answer will always be **positive.**

**Odd powers of negative numbers,** however, always leave one factor of the negative number not paired. This one lone negative term guarantees that the answer will always be **negative.**

**Zero Exponents**

The number zero may be used as an exponent.

**The value of any expression raised to the zero power is 1.**

(Except zero raised to the zero power is undefined.)

**Negative Exponents**

Negative numbers as exponents have a special meaning.

The rule is as follows:

**For example:**

**Exponents and Units**

When working with units and exponents (or powers), remember to adjust the units appropriately.

(36 ft)^{3 }= (36 ft) • (36 ft) • (36 ft)

= (36 • 36 • 36) (ft • ft • ft)

= 46656 ft^{3}

Exponents can be very useful for evaluating expressions. It is also useful to learn how to use your calculator when working with exponents