How do you simplify expressions with negative bases using the product rule?

When simplifying expressions with negative bases using the product rule, the process is the same as with positive bases: you add the exponents if the bases are identical. For example, consider -3 5 × -3 2 . Since both have the base -3 , you add the exponents: 5 + 2 = 7, resulting in -3 7 . The negative sign is part of the base and remains unchanged. This rule helps simplify complex expressions involving negative bases raised to powers.

6. Exponents and Polynomials

The Product Rule

6. Exponents and Polynomials

The Product Rule: Videos & Practice Problems

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Topic summary

Understanding exponential expressions involves recognizing that exponents represent repeated multiplication of a base. When multiplying two 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 aids in simplifying polynomials and monomials, reinforcing the importance of identifying the base and exponent in exponential expressions for efficient algebraic manipulation.

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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.

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Problem

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

$12 to the sixth power$

$12 to the 12th power$

$12 to the ninth power$

$12 to the 27th power$

Problem

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

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

$4 to the seventh power$

${(- 4)}^{7}$

${(- 4)}^{8}$

Problem

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

$3 x cubed y$

$3 x y cubed$

$3 x cubed$

$3 x y squared$

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

$6 s to the seventh power t$

$5 s to the seventh power t cubed$

$6 s to the seventh power t cubed$

$5 s cubed t to the seventh power$

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.

Here’s what students ask on this topic:

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. For example, $4_{2} \times 4_{1} = 4_{3}$ because you add 2 and 1 to get 3. This rule simplifies expressions by reducing the multiplication of powers to a single power with an exponent equal to the sum of the original exponents.

When simplifying expressions with negative bases using the product rule, the process is the same as with positive bases: you add the exponents if the bases are identical. For example, consider ${-3}_{5} \times {-3}_{2}$ . Since both have the base $-3$ , you add the exponents: 5 + 2 = 7, resulting in ${-3}_{7}$ . The negative sign is part of the base and remains unchanged. This rule helps simplify complex expressions involving negative bases raised to powers.

We add exponents when multiplying powers 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 expressions, you are essentially multiplying $a$ by itself $m + n$ times. Therefore, the exponents add up, which is why the product rule is $a_{m} \times a_{n} = a_{m + n}$ .

To apply the product rule to variables with large exponents, you follow the same principle: add the exponents if the bases are the same. For example, if you have $x_{30} \times x_{70}$ , since both bases are $x$ , you add the exponents 30 + 70 to get 100. The simplified expression is $x_{100}$ . This rule works regardless of how large the exponents are, making it easier to handle expressions with very large powers.

In algebra, the 'x' symbol and the dot symbol both represent multiplication, but they are used differently to avoid confusion. The 'x' symbol is often used for multiplication but can be confused with the variable $x$ . To prevent this, the dot symbol $⋅$ is commonly used for multiplication, especially when variables like $x$ are involved. For example, $x ⋅ y$ clearly shows multiplication between variables without confusion. Both symbols mean the same operation, but the dot is preferred in algebraic expressions to maintain clarity.