## Basic Exponent Laws II

A monomial consists of coefficients (numbers) and variables (letters). The variables may have exponents.  For the monomial:
$$6x^5y^3$$

the coefficient is $6$, the variables are $x$ and $y$.

#### Law of Multiplication

When multiplying powers with the same base, keep the base and add the exponents.
$$x^5 \times x^2=x^7$$

When multiplying monomials with coefficients, multiply the coefficients normally.
$$6x^5 \times 3x^2=18x^7$$

When there is more than one variable, group common variables together and multiply.
$$6x^5y^3 \times 3x^2y^{-1}=18x^7y^2$$

Common Mistake: There are no rules to addition. Evaluate any power individually.

$2^3+2^2 \neq 2^5=32$

$2^3+2^2 =8+4=12$

#### Law of Division

When dividing powers with the same base, keep the base and subtract the exponents.
$$x^5 \div x^2=x^3$$

Always subtract the power with the larger exponent by the power with the smaller exponent. So if the exponent in the denominator term is higher, place the answer in the denominator. A 1 should be written if the numerator is left blank.
$$\frac{x^2}{x^5}=\frac{1}{x^3}$$

Reduce any coefficients similar to the way you would reduce fractions:
$$\frac{12x^2}{16x^5}=\frac{3}{4x^3}$$

When there is more than one variable, group common variables together and divide.
$$\frac{12x^2y^{-3}}{16x^5y^{-5}}=\frac{3y^2}{4x^3}$$

If the exponents are equal on two powers with the same base, the powers will cancel each other or reduce to 1.
$$\frac{12x^2y^{-3}z^4}{16x^5y^{-5}z^4}=\frac{3y^2}{4x^3}$$

#### Law of Powers

When raising a powers to a higher exponent, keep the base and multiply the exponents.
$$(x^5)^2=x^{10}$$

When the coefficient is inside the brackets, the exponent must apply to the coefficient as well. Be careful though: You don’t multiply the exponent to the coefficient!

$(3x^5)^2 \neq 6x^{10}$

$(3x^5)^2=3^2 \cdot (x^5)^2=9x^{10}$

If the coefficient is NOT inside the brackets, the exponent does not apply to the coefficient. $$3(x^5)^2=3x^{10}$$

### Practice

EXAMPLE 1

Simplify then evaluate: $2^{11} \div 2^8$

###### SOLUTION

Using the Law of Division: $$2^{11} \div 2^8=2^{11-8}=2^3$$

Evaluating: $2^3=8$

EXAMPLE 2

Simplify: $(m^0)^{10}$

###### SOLUTION

Using the Law of Powers: $$(m^0)^{10}=m^0$$

We should more properly use the Law of Zero Exponents to write: $m^0=1$

EXAMPLE 3

Find the value of $x$ that makes the equation true: $4^x \times 4^4=4^7$

###### SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: $$x+4=7$$

Therefore: $x=3$