4.3 SAT Math - Exponents

An exponent represents the number of times something is multiplied by itself.

In the example above, x⁶means x has been multiplied by itself 6 times, or x•x•x•x•x•x.  The coefficient of 5 means we are dealing with five of those groupings.
The easiest way to work with exponents, especially on the SAT, is to remember that most exponents are one step "behind" regular numbers.  Here's how that works:

Adding/Subtracting

Must have: Same base AND same exponent (for example, x² + 2x²)

Operation: Add or subtract the COEFFICIENTS.  Do NOTHING to the base or exponent.

a³ - a² = not possible
a³ + a³ = 2a³
2a⁴ + a⁴ = 3a⁴

How is this one step 'behind'?  If you're adding regular numbers, you're doing nothing to the exponent.

Multiply/divide

Must have: Same base (y³•y⁴)
Operation: Multiply any coefficients, add the exponents; or divide the coefficients, subtract the exponents.

x²•x³ = x⁵
3x³•x⁴ = 3x⁷

One step behind: If you're multiplying regular numbers, you're adding exponents.

Raising to a power

Must have: Things being raised to a power, usually with parentheses (for example, (4w³xz²)⁵)
Operation: Raise any coefficients to the outside power, multiply the exponents.

(b³)⁴ = b¹²
(5m⁴n)³ = 125m¹²n³

-SAT Math Tip:  Watch the placement of parentheses. Anything inside the parentheses gets raised to the outside power:

4(b²)³ = 4b⁶
(4b²)³ = 64b⁶

One step behind: If you are raising regular numbers to a power, you are multiplying exponents.
-Hint: Remember, your base never changes!  If your base is a variable, that's easy to remember, but what about when you have a number?

2³•2⁴ = 2⁷ NOT 4⁷
                                                        
Keep in mind the rules for negatives: any two negatives multiplied together become positive, so any negative number raised to an even numbered power will be positive:

(-3)⁶ = +729

Likewise, any negative number raised to an odd-numbered power will be negative:

(-3)⁵ = -243

When raising fractions to a power, remember that you can split the numerator and denominator to make it easier if you like:

(1/4)³ = 1³/4³

But more importantly, fractions less than 1 raised to a power greater than 1 will in fact become smaller because essentially you're dividing your object into that many more pieces.  So if our fraction were a pizza, and we divided it into fourths, once you cube that number, you're dividing it into sixty-fourths instead.

(1/4)³ = 1/64