The Product Rule: Videos & Practice Problems

Understanding exponential expressions involves recognizing that exponents represent repeated multiplication of a base. When multiplying exponential expressions with the same base, the product rule applies: add the exponents to simplify, as in $a^m\times a^n=a^{m+n}$. This rule is essential for simplifying polynomials and monomials, especially when dealing with terms involving powers and coefficients. Recognizing the base and correctly applying exponent rules enhances skills in manipulating algebraic expressions and understanding polynomial degrees.

The Product Rule for Exponents

Exponents represent repeated multiplication, where a number raised to a power indicates how many times it is multiplied by itself. For example, three squared means multiplying three by itself twice: 3 × 3. When working with expressions involving exponents, especially when multiplying or dividing them, specific rules help simplify these operations efficiently.

One fundamental rule is the product rule for exponents. This rule applies when multiplying two exponential expressions that share the same base. Instead of multiplying the bases repeatedly, you add the exponents. For instance, multiplying 4² by 4¹ is equivalent to 4 raised to the power of 2 + 1, which equals 4³. This works because 4² × 4¹ expands to 4 × 4 × 4, or 4 multiplied by itself three times.

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.

For example, consider multiplying (-3)⁵ by (-3)². Since the base is the same (-3), add the exponents: 5 + 2 = 7. The simplified expression is (-3)⁷, which evaluates to -2,871.

Another example is multiplying x³⁰ by x⁷⁰. Both have the base x, so add the exponents: 30 + 70 = 100. The simplified form is x¹⁰⁰.

It is important to recognize the symbols used for multiplication. The letter "x" can represent a variable, so a dot (·) is often used to indicate multiplication to avoid confusion. Regardless of the symbol, the product rule remains the same: when multiplying exponential expressions with the same base, add the exponents.

Understanding and applying the product rule for exponents allows for efficient simplification and manipulation of exponential expressions, which is essential in algebra and higher-level mathematics.

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

• A. $12^6$
• B. $12^{12}$
• C. $12^9$
• D. $12^{27}$

Answer: (B)

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

• A. $\left(-4\right)^5$
• B. $4^7$
• C. $\left(-4\right)^7$
• D. $\left(-4\right)^8$

Answer: (C)

Simplify each expression.
$xy\cdot 3x^2$

• A. $3x^3y$
• B. $3xy^3$
• C. $3x^3$
• D. $3xy^2$

Answer: (A)

Simplify each expression.
$(-5a^2)\left(3a^8\right)$

• A. $-2a^{10}$
• B. $-15a^6$
• C. $-5a^{10}$
• D. $-15a^{10}$

Answer: (D)

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

• A. $6s^7t$
• B. $5s^7t^3$
• C. $6s^7t^3$
• D. $5s^3t^7$

Answer: (C)

The Product Rule for Exponents Example 1

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.

Here’s what students ask on this topic:

What is the product rule for exponents and how do you apply it?

The product rule for exponents states that when multiplying two exponential expressions with the same base, you add the exponents. Mathematically, this is expressed as $a_m\times a_n=a_{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents. To apply this rule, first confirm that the bases are the same. Then, simply add the exponents together and keep the base unchanged. For example, $4_2\times 4_3=4_5$. This rule simplifies expressions and helps in solving algebraic problems involving powers.

Can the product rule be used if the bases are different?

No, the product rule for exponents only applies when the bases of the exponential expressions are the same. If the bases are different, you cannot add the exponents directly. For example, $2_3\times 3_4$ cannot be simplified using the product rule because the bases 2 and 3 are different. In such cases, you either leave the expression as is or use other algebraic methods depending on the problem context. Always check the base first before applying the product rule.

How do you simplify expressions like $x_{30}\times x_{70}$ using the product rule?

To simplify $x_{30}\times x_{70}$ using the product rule, first confirm that the bases are the same, which they are (both are $x$). Then, add the exponents: 30 + 70 = 100. So, the expression simplifies to $x_{100}$. This is the simplest form of the product of these two exponential expressions. This method works for any base and exponents as long as the base is the same.

Why do we add exponents when multiplying exponential expressions with the same base?

We add exponents when multiplying exponential expressions with the same base because exponents represent repeated multiplication. For example, $a_m$ means multiplying $a$ by itself $m$ times, and $a_n$ means multiplying $a$ by itself $n$ times. When you multiply these two expressions, you are essentially multiplying $a$ by itself $m+n$ times. Therefore, the exponents add up, which is why the product rule is $a_m\times a_n=a_{m+n}$.

How do you handle negative bases when using the product rule?

When using the product rule with negative bases, the process of adding exponents remains the same as long as the bases are identical, including the negative sign. For example, consider ${-3}_5\times {-3}_2$. Since both bases are $-3$, you add the exponents: 5 + 2 = 7. The expression simplifies to ${-3}_7$. However, be cautious when evaluating the expression numerically, as the negative base raised to an odd or even power affects the sign of the result. The product rule itself only concerns the exponents and base, not the sign of the final value.

What is the difference between the multiplication symbols used in exponential expressions?

In exponential expressions, multiplication can be represented by different symbols, commonly the '×' symbol or a dot '·'. For example, $x_2\times x_3$ and $x_2·x_3$ both mean the same thing: multiply $x_2$ by $x_3$. The dot is often preferred in algebra to avoid confusion with the variable 'x'. Regardless of the symbol used, the product rule applies the same way: add the exponents when the bases are the same.