Problem 2 - Precalculus Mathematics for Calculus

Question

(a) When we multiply two powers with the same base, we __________ the exponents. So \(3^{4} \cdot 3^{5}=\) __________. (b) When we divide two powers with the same base, we __________ the exponents. So \(\frac{3^{5}}{3^{2}}=\) __________.

Step-by-Step Solution

Answer

(a) Add; 3^9. (b) Subtract; 3^3.

Step 1: Understanding Multiplication of Powers

When multiplying two powers that have the same base, we add the exponents. For instance, for two powers \(3^4\) and \(3^5\), we keep the base (3) and add the exponents: \(4 + 5 = 9\).

Step 2: Calculate Multiplication Example

Using the rule from Step 1, calculate \(3^4 \cdot 3^5 = 3^{4+5} = 3^9\). Thus, the product of the powers is \(3^9\).

Step 3: Understanding Division of Powers

When dividing two powers with the same base, we subtract the exponent of the denominator from the exponent of the numerator. For example, in \(\frac{3^5}{3^2}\), keep the base (3) and subtract the exponents: \(5 - 2 = 3\).

Step 4: Calculate Division Example

Using the rule from Step 3, calculate \(\frac{3^5}{3^2} = 3^{5-2} = 3^3\). Thus, the result of the division is \(3^3\).

Key Concepts

Multiplication of Powers

When dealing with exponents, if two powers share the same base, the operation becomes straightforward. During multiplication, you simply add the exponents.
For example, if the powers are \(3^4\) and \(3^5\), they both have the base of 3.
The multiplication rule for exponents tells us to add the exponents: \(4 + 5 = 9\). Therefore, \(3^4 \cdot 3^5\) simplifies to \(3^9\).

• Rule: If \(a^m \cdot a^n\), then \(a^{m+n}\).
• Example: \(3^2 \cdot 3^3 = 3^{2+3} = 3^5\).
• Note: Always ensure the base is the same when using this rule!

Division of Powers

In a similar way to multiplication, dividing powers with the same base involves combining their exponents, but instead of adding, you subtract them.
The rule is: keep the base and subtract the exponent of the denominator from the exponent of the numerator. For example, with \(\frac{3^5}{3^2}\), the base remains as 3.
The exponent from the denominator (2) is subtracted from the exponent in the numerator (5): \(5 - 2 = 3\). Thus, \(\frac{3^5}{3^2}\) reduces to \(3^3\).

• Rule: If \(\frac{a^m}{a^n}\), then \(a^{m-n}\).
• Example: \(\frac{2^4}{2^2} = 2^{4-2} = 2^2\).
• Remember: The base must be the same to apply this rule!

Same Base

Understanding that the base of the power is crucial is key to applying these exponent rules.
Exponents are convenient for representing repeated multiplication of the same number. When we talk about the 'base', we're referring to this repeated factor.
In our examples like \(3^4\) or \(3^5\), 3 is the base, indicating multiplication of 3 by itself multiple times.

• Crucial Concept: Same base allows us to perform arithmetic operations directly on the exponents.
• Tip: Always identify the base first to determine if rules can be applied.
• Example: In \(5^3 \cdot 5^2\) or \(\frac{5^5}{5^3}\), the base '5' remains constant through calculations.