11. Exponents and Polynomials

The Product Rule

11. Exponents and Polynomials

The Product Rule: Videos & Practice Problems

Video Lessons

Concept

The Product Rule for Exponents

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The Product Rule for Exponents Video Summary

Exponents represent repeated multiplication, where an expression like three squared means multiplying three by itself twice. When working with exponential expressions, especially when multiplying or dividing them, specific rules help simplify and manipulate these expressions efficiently. One fundamental rule is the product rule for exponents, which applies when multiplying two exponential expressions with the same base.

For example, consider the expression $4^2 \times 4^1$. Since \$4^2$ means 4 multiplied by itself twice and $4^1$ means 4 once, multiplying these together is equivalent to \(4 \times 4 \times 4$, or \)4^3\(. This shows that when multiplying exponential expressions with the same base, you add the exponents. Mathematically, the product rule is expressed as:

\[a^m \times a^n = a^{m+n}\]

where a is the base, and m and n are the exponents. A helpful way to remember this rule is by noting that both multiplication and addition symbols resemble crosses, linking the operation of multiplying expressions to adding their exponents.

For instance, if you multiply \)(-3)^5$ by $(-3)^2$, since the base is the same (-3), you add the exponents: \(5 + 2 = 7$. So, the expression simplifies to \)(-3)^7\(, which evaluates to -2,871.

Similarly, multiplying \)x^{30}$ by $x^{70}$ results in $x^{100}$ by adding the exponents 30 and 70. It's important to recognize that multiplication can be denoted by different symbols, such as the dot (·) or the letter x. The dot is often preferred in algebra to avoid confusion with the variable x.

Understanding and applying the product rule for exponents is essential for simplifying expressions and solving algebraic problems involving powers. By consistently adding exponents when multiplying like bases, you can efficiently rewrite and evaluate exponential expressions.

Problem

Simplify each expression using the product rule if possible.
$12^{3} \cdot 12^{9}$

$12 to the sixth power$

$12 to the 12th power$

$12^9$

$12^{27}$

Problem

Simplify each expression using the product rule if possible.
$(- 4)^{6} (- 4)$

${(- 4)}^{5}$

$4 to the seventh power$

$$\left$(-4$\right$)^7$

$$\left$(-4$\right$)^8$

Problem

Simplify each expression.
$x y times 3 x squared$

$3x^3y$

$3xy^3$

$3x^3$

$3xy^2$

Problem

Simplify each expression.
$(- 5 a^{2}) (3 a^{8})$

$- 2 a^{10}$

$- 15 a^{6}$

$- 5 a^{10}$

$- 15 a^{10}$

Problem

Simplify each expression.
$(2 s^{3} t) (3 s^{4} t^{2})$

$6 s to the seventh power t$

$5 s to the seventh power t cubed$

$6 s to the seventh power t cubed$

$5s^3t^7$

Example

The Product Rule for Exponents Example 1

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The Product Rule for Exponents Example 1 Video Summary

To find the area of a square with a side length of 5x³ centimeters, we start by recognizing that all sides of a square are equal. Therefore, each side measures 5x³. The formula for the area of a square is the product of its two side lengths, which means we multiply 5x³ by itself.

When multiplying, it helps to group like terms: multiply the constants and the variable parts separately. Multiplying the constants, 5 × 5, gives 25. For the variable part, x³ × x³, we apply the product rule of exponents, which states that when multiplying powers with the same base, we add the exponents. Thus, x³ × x³ = x^{3+3} = x^6.

Combining these results, the area of the square is expressed as:

\[\text{Area} = 25x^6 \text{ cm}^2\]

This expression represents the simplest form of the area, showing how algebraic expressions and exponent rules are applied to geometric problems involving polynomial side lengths.