Problem 159 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. In parts (a) and (b), complete each statement. a. \(b^{4} \cdot b^{3}=(b \cdot b \cdot b \cdot b)(b \cdot b \cdot b)=b^{2}\) b. \(b^{5} \cdot b^{5}=(b \cdot b \cdot b \cdot b \cdot b)(b \cdot b \cdot b \cdot b \cdot b)=b^{?}\) c. Generalizing from parts (a) and (b), what should be done with the exponents when multiplying exponential expressions with the same base?

Step-by-Step Solution

Answer

The general rule when multiplying exponents with the same base can be observed with these examples: the exponents are added together. Therefore, \(b^{4} \cdot b^{3}=b^{7}\) and \(b^{5} \cdot b^{5}=b^{10}\).

Step 1: Identify Error in Given Statement

Looking closer to the first statement: \(b^{4} \cdot b^{3}=(b \cdot b \cdot b \cdot b)(b \cdot b \cdot b)=b^{2}\), one can see there's a mistake in the final expression. According to the multiplication property of exponents, \(b^{4} \cdot b^{3}\) should give \(b^{7}\), not \(b^{2}\).

Step 2: Evaluate \(b^{5} \cdot b^{5}\)

The multiplication property of exponents can be applied: \(b^{5} \cdot b^{5}=(b \cdot b \cdot b \cdot b \cdot b)(b \cdot b \cdot b \cdot b \cdot b)\). When the bases are the same (in this case, \(b\)), the exponents are added together, resulting in \(b^{10}\), not \(b^{?}\). So, the correct statement should be \(b^{5} \cdot b^{5}=b^{10}\).

Step 3: Generalizing the Rule of Exponential Multiplication

Based on the evaluations of part (a) and (b), it can be concluded that when multiplying exponential expressions with the same base, the exponents should be added together.

Key Concepts

Multiplication Property of Exponents

The multiplication property of exponents is central to simplifying expressions where two exponents have the same base. If you have two exponential expressions like \( a^m \) and \( a^n \), the multiplication property tells us that we can simply add the exponents when multiplying these expressions. This transforms the expression into \( a^{m+n} \).
This property applies whenever you multiply terms that are exponential and share a common base. For example, in the expression \( b^4 \cdot b^3 \), both terms have the base "b". According to the multiplication property of exponents, you combine \( b^4 \) and \( b^3 \) by adding the exponents resulting in \( b^{4+3} = b^7 \).

• The bases must be the same for this property to apply.
• This property helps simplify larger expressions and solve more complex equations.

Remember, this rule does not apply to expressions with different bases.

Exponential Expressions

Exponential expressions are mathematical expressions that involve numbers raised to a power or exponent. An expression like \( b^n \) shows an exponential form where "b" is the base, and "n" is the exponent.
Exponents indicate how many times the base is multiplied by itself. For example, in \( b^3 \), the base "b" is used as a factor three times: \( b \times b \times b \).

• Exponential expressions are a compact way to represent large calculations.
• They are very helpful in mathematical fields like algebra and calculus.

Using exponential expressions helps simplify calculations and expressions that involve repeated multiplication.

Same Base Exponent Rule

The same base exponent rule is an easy way to simplify calculations involving exponentials. When you multiply exponential expressions that share the same base, you add their exponents together. Let's consider an example: \( b^5 \cdot b^5 \). Here, both terms share the base "b".
According to the same base exponent rule, the expression becomes \( b^{5+5} \), which simplifies to \( b^{10} \).

• The rule simplifies multiplications into simple addition, making it much easier to work with exponents.
• It can greatly reduce the complexity of algebraic computations.

The rule is extremely useful in algebra where handling powers efficiently is crucial to problem-solving.