Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. \(\left(3^{3} \div 3^{4}\right)^{5}\)

Step-by-Step Solution

Answer

The expression is \( \frac{1}{3^5} \) with a positive exponent.

Step 1: Apply the Quotient Rule

The expression given is \( \left(3^{3} \div 3^{4}\right)^{5} \). Start by using the Quotient Rule for exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \). Apply this to the base: \( 3^{3 - 4} = 3^{-1} \). Now, the expression is \( \left(3^{-1}\right)^{5} \).

Step 2: Apply the Power Rule

Use the Power Rule for exponents, which tells us that \( (a^m)^n = a^{m \cdot n} \). Apply this rule to \( \left(3^{-1}\right)^{5} \), which results in the base being raised to the product of the exponents: \( 3^{-1 \times 5} = 3^{-5} \).

Step 3: Convert Negative Exponent to Positive

To write the expression with a positive exponent, use the negative exponent rule \( a^{-m} = \frac{1}{a^m} \). Apply this to the result from Step 2: \( 3^{-5} = \frac{1}{3^5} \).

Key Concepts

Quotient Rule

When dividing exponents with the same base, you can simplify the expression using the Quotient Rule. This rule states that \( \frac{a^m}{a^n} = a^{m-n} \). This handy rule helps to combine terms with exponents into a single term. In our exercise, the expression is \( \left(3^{3} \div 3^{4}\right)^{5} \). We need to start by applying the Quotient Rule to combine the base exponents:

• Identify the exponents: \(3\) and \(4\)
• Subtract them: \(3 - 4 = -1\)
• Rewrite the expression: \(3^{-1}\)

Now the expression inside the parentheses is simplified to \(3^{-1}\). Using this rule simplifies the expression before raising it to the power indicated outside the parentheses, making the overall calculation easier.

Power Rule

The Power Rule is essential when dealing with exponents raised to another power. According to this rule, \((a^m)^n = a^{m \cdot n}\). It allows us to multiply the exponents when a power is raised to another power. In our context, the expression we obtained from the Quotient Rule is \((3^{-1})^5\).

• Identify the base: \(3\)
• Multiply the exponents: \(-1 \times 5 = -5\)
• Write the expression: \(3^{-5}\)

Using the Power Rule streamlines the simplification process by condensing multiple exponent operations into a single term. This step is crucial to reach a simplified form that incorporates both the base and the power raised without any additional complexity.

Negative Exponents

Negative exponents may seem tricky at first, but they're quite simple once you understand their meaning. A negative exponent indicates that the base should be put into the denominator and transformed to a reciprocal. The rule is \(a^{-m} = \frac{1}{a^m}\). This conversion results in a positive exponent, which is often preferred for final expressions.In our original problem, after using the Power Rule, we reach \(3^{-5}\). To express this with a positive exponent:

• Use the negative exponent rule: \(3^{-5} = \frac{1}{3^5}\)

This formalizes the expression into a positive exponent format, making it easier to understand and use in further calculations. Remember, dealing with negative exponents is simply about flipping the base to the other side of the fraction bar and changing the sign of the exponent.