Problem 20 - College Algebra

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

\(\frac{1}{3^5}\)

Step 1: Apply the Quotient Rule for Exponents

According to the quotient rule for exponents, \( a^m \div a^n = a^{m-n} \). Apply this rule to the expression \( 3^3 \div 3^4 \).\[ 3^3 \div 3^4 = 3^{3-4} = 3^{-1} \]

Step 2: Evaluate the Power of a Power

Apply the power of a power rule which states \( (a^m)^n = a^{m \cdot n} \) to the expression \( (3^{-1})^5 \).\[ (3^{-1})^5 = 3^{-1 \cdot 5} = 3^{-5} \]

Step 3: Convert to Positive Exponent

Rewrite \( 3^{-5} \) with a positive exponent by using the property \( a^{-n} = \frac{1}{a^n} \).\[ 3^{-5} = \frac{1}{3^5} \]

Key Concepts

Exponent Rules

Exponent rules are essential for simplifying algebraic expressions involving powers of numbers. These rules help us manipulate and simplify expressions with exponents.
They are foundational in algebraic calculations.Some of the key exponent rules include:

• **Product Rule**: When multiplying like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).
• **Quotient Rule**: When dividing like bases, subtract the exponents: \( a^m \div a^n = a^{m-n} \).
• **Power of a Power Rule**: To elevate a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• **Negative Exponent Rule**: A negative exponent indicates the reciprocal: \( a^{-n} = \frac{1}{a^n} \).
• **Zero Exponent Rule**: Any number raised to the zero power is one: \( a^0 = 1 \) (given \( a eq 0 \)).

Understanding these rules allows you to approach problems systematically, ensuring that expressions are expressed in their simplest form. This simplification is particularly valuable when complex calculations arise.

Quotient Rule for Exponents

The quotient rule for exponents simplifies expressions that involve division of like bases with different exponents. It can make complex divisions much simpler to handle.This rule states that when you divide two powers with the same base, you simply subtract the exponent of the denominator from the exponent of the numerator: \( a^m \div a^n = a^{m-n} \).**Practical Application**:
Let's say you have the expression \( 3^3 \div 3^4 \). Using the quotient rule, subtract the exponents:\[3^3 \div 3^4 = 3^{3-4} = 3^{-1}\]This tells us that instead of performing direct division, we can simply change the exponent, resulting in a more straightforward expression.The quotient rule is a powerful tool in algebra which makes it easier to work with polynomial expressions and functions involving exponentiation.

Power of a Power Rule

The power of a power rule in exponents is a fundamental concept where a power is raised to another power. This rule allows us to simplify expressions efficiently by combining the exponents.According to this rule, when you have a power raised to another power, you multiply the exponents:\((a^m)^n = a^{m \cdot n}\).**Example**:
Consider evaluating \( (3^{-1})^5 \).- With the power of a power rule, you multiply the exponents: \[3^{-1 \cdot 5} = 3^{-5}.\]Instead of calculating the power repeatedly, simply multiply the exponents to simplify the expression.This method of simplifying helps keep expressions tidy and manageable, especially in more advanced algebraic problems. Ensuring mastery of this concept early makes later stages of learning algebra less daunting and more intuitive.