Chapter 8: Variables and Expressions 105When you work with exponents, there are three basic rules to remember.
Multiplication: When you multiply powers of the same base, keep the base and add the exponents.
This means that x^2 x^3 = x^5 because x^2 x^3 = (xx)(xxx) = x^5. Don’t forget that when you have a
single variable, like y, even though you don’t see an exponent, y is y^1. Multiplying yy^3 gives you
y^4. Multiplying a power of a variable by the variable again raises the exponent.
Division: When you divide powers of the same base, keep the base and subtract the exponents.
This means that x
x
x8
6
^2. Don’t forget that a variable to the zero power equals 1.
Powers: When you raise a power to a power, keep the base and multiply the exponents.
This one doesn’t come up as often, but it means that (t^7 )^3 = t^21. Remember that exponents work
only on what they touch, so 3x^2 means 3xx, because the exponent is only touching the x, but
(3x)^2 = (3x)(3x) because the exponent is touching the parenthesis, and so the whole quantity is
squared.Ways to Multiply
You can multiply a constant by a variable or a power of a variable: -5x^2 = -5x^2.
You can multiply a constant by a product of a constant and a variable: 13 4 t = 52t.
You can multiply a variable or a power of a variable by a variable or a power of a variable: yy^3 = y^4.
You can multiply a variable or a power of a variable by a product of a constant and a variable or a
power of a variable.: a^2 6 a^3 = 6a^5.
You can multiply the product of a constant and a variable (or a power of a variable) by a product
of a constant and a variable (or a power of a variable): -8x^2 4 x^2 = -32x^4.
And you can even multiply a constant, a variable, or the product of a constant and a
variable by an expression: 2x(3x – 5) = 2x(3x) – 2x(5) = 6x^2 – 10x and 3t^2 (t^2 + 2t + 1) =
3 t^2 (t^2 ) + 3t^2 (2t) + 3t^2 (1) = 3t^4 + 6t^3 + 3t^2.
In case you were wondering, you can multiply with two different variables. You just simplify as
much as possible: 5x(2y) = 10xy and 2ab^2 (3a^2 b) = 2 3 aa^2 b^2 b = 6a^3 b^3.CHECK POINT
Put each product in simplest form.- 5(-6a)
- x(4x^2 )
- 2y(-5y^3 )
9. 3t^2 (-8t^5 )
10. 7z(3z + 5)