6.2 Use Multiplication Properties of Exponents

Exponent Properties

Exponents are shorthand for repeated multiplication. Instead of writing $2 \times 2 \times 2$, you write $2^3$. The properties covered here give you rules for simplifying expressions that involve exponents, which you'll use constantly when working with polynomials.

Simplification with exponent properties

Three core properties let you simplify most exponent expressions:

• Product property: When you multiply terms with the same base, add the exponents. $a^m \times a^n = a^{m+n}$
• Power property: When you raise a power to another power, multiply the exponents. $(a^m)^n = a^{m \times n}$
• Product to power property: When a product inside parentheses is raised to a power, distribute that exponent to every factor inside. $(a \times b)^n = a^n \times b^n$

Each of these comes from the definition of exponents as repeated multiplication. The sections below show how each one works.

Product property of exponents

When two terms share the same base, you can combine them by adding exponents:

$x^3 \times x^4 = x^{3+4} = x^7$

Why does this work? $x^3$ means $x \times x \times x$ and $x^4$ means $x \times x \times x \times x$. Multiply them together and you get seven $x$'s multiplied, which is $x^7$.

The bases must match for this to work. You can't combine $2^3 \times 3^2$ using the product property because the bases (2 and 3) are different. You'd just compute each one separately: $8 \times 9 = 72$.

Power property of exponents

When a power is raised to another power, multiply the exponents:

$(y^2)^3 = y^{2 \times 3} = y^6$

Think of it this way: $(y^2)^3$ means $y^2 \times y^2 \times y^2$. Now use the product property to add the exponents: $2 + 2 + 2 = 6$.

This also works when the base is more complex. For example:

$((2x)^2)^3 = (2x)^{2 \times 3} = (2x)^6$

To finish simplifying $(2x)^6$, you'd then use the product to power property (next section) to get $2^6 \times x^6 = 64x^6$.

Product to power property

When a product inside parentheses is raised to a power, distribute the exponent to each factor:

$(2x)^3 = 2^3 \times x^3 = 8x^3$

Each factor inside the parentheses gets raised to the power separately. If there are more than two factors, the same idea applies:

$(3ab)^2 = 3^2 \times a^2 \times b^2 = 9a^2b^2$

A common mistake is forgetting to apply the exponent to the numerical coefficient. In $(2x)^3$, the 2 also gets cubed. $2^3 = 8$, not 2.

Laws of Exponents

Two additional rules handle special cases:

• Zero exponent: Any nonzero number raised to the 0 power equals 1. $a^0 = 1$ (as long as $a \neq 0$). For example, $5^0 = 1$ and $(3x)^0 = 1$.
• Negative exponents: A negative exponent means "take the reciprocal." $a^{-n} = \frac{1}{a^n}$. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Scientific notation uses these ideas to express very large or very small numbers as a number between 1 and 10 multiplied by a power of 10. For example, $3{,}500{,}000 = 3.5 \times 10^6$ and $0.004 = 4 \times 10^{-3}$.

Combining exponent properties

Most real problems require more than one property. Here's a step-by-step example:

1. Apply the product to power property to each set of parentheses separately:

   • $(3x^2y)^3 = 3^3 \times (x^2)^3 \times y^3 = 27x^6y^3$
   • $(2xy^2)^2 = 2^2 \times x^2 \times (y^2)^2 = 4x^2y^4$

2. Multiply the results using the product property (add exponents of matching bases):

   • $27x^6y^3 \times 4x^2y^4 = (27 \times 4) \times x^{6+2} \times y^{3+4} = 108x^8y^7$

The strategy is always the same: simplify inside parentheses first, then combine like bases.

Monomial multiplication using exponents

A monomial is a single-term expression like $4x^3y^2$. To multiply two monomials:

1. Multiply the numerical coefficients.
2. For each variable, add the exponents.

Example: $(4x^3y^2) \times (2xy^3)$

• Coefficients: $4 \times 2 = 8$
• $x$: $x^3 \times x^1 = x^{3+1} = x^4$
• $y$: $y^2 \times y^3 = y^{2+3} = y^5$
• Result: $8x^4y^5$

Another example: $(12x^2y) \times (3xy^2)$

• Coefficients: $12 \times 3 = 36$
• $x$: $x^2 \times x^1 = x^3$
• $y$: $y^1 \times y^2 = y^3$
• Result: $36x^3y^3$

Notice that when a variable appears without a written exponent (like $x$), the exponent is 1.