Problem 29 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\frac{4^{-2} \cdot 4^{-1}}{4^{-3}}$$

Step-by-Step Solution

Answer

The answer is 1.

Step 1: Simplify the Numerator

The first step is to combine the exponents in the numerator. We have the terms \(4^{-2}\) and \(4^{-1}\). To simplify, add the exponents: \[4^{-2} \cdot 4^{-1} = 4^{-2 + (-1)} = 4^{-3}\].

Step 2: Apply the Quotient Rule

Now, we have a fraction with a single base: \[\frac{4^{-3}}{4^{-3}}\]The quotient rule states that when dividing like bases with exponents, we subtract the exponents: \[4^{-3 - (-3)} = 4^{0}\].

Step 3: Simplify the Power of Zero

Any non-zero base raised to the power of zero equals 1. Thus, \(4^{0} = 1\).

Key Concepts

Negative Exponents

Understanding negative exponents is crucial when working with algebraic expressions. A negative exponent indicates that the base is on the wrong side of the fraction line and needs to be flipped. For example, when you see a negative exponent like in the expression \(4^{-2}\), it means \(\frac{1}{4^2}\). This can be rewritten by taking the reciprocal: \(\frac{1}{16}\).
Negative exponents essentially represent division. Instead of multiplying the base by itself, you are dividing 1 by that base multiple times. This concept flips the base across the fraction, changing multiplication into division to simplify expressions with negative exponents.
So, the next time you encounter a negative exponent, remember: it's just a signal to flip the base from the numerator to the denominator, or vice versa.

Quotient Rule for Exponents

The quotient rule for exponents is a fundamental tool for simplifying expressions involving division of like bases with exponents. The rule states that when you divide two exponents with the same base, you can subtract the exponent in the denominator from the exponent in the numerator.
For example, if you have an expression like \( \frac{a^m}{a^n} \), the quotient rule lets you rewrite this as \( a^{m-n} \).

• This rule helps simplify expressions by reducing the complex numbers into a single exponent.
• It's particularly useful when dealing with large numbers or multiple variables with exponents.
  

Applying this concept to step 2 from our original solution, we used the rule to simplify the expression \(\frac{4^{-3}}{4^{-3}}\) as \(4^{0}\), since \(-3 - (-3) = 0 \). This ultimately simplifies the expression to 1, as any number raised to the zero power is 1.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing the complexity of expressions while keeping the original value unchanged. This process includes combining like terms, factoring, and applying exponent rules effectively. A simplified expression is easier to read, understand, and solve.
With the original problem, we started by simplifying the numerator using negative exponent rules and combined exponents by adding them: \((4^{-2} \cdot 4^{-1} = 4^{-3})\).

• Next, we applied the quotient rule for exponents to simplify the fraction resulting in \(4^{0}\).
• Lastly, we simplified \(4^{0}\) to 1, knowing that any number raised to 0 equals one.

By understanding and applying these rules correctly, you can approach even complicated algebraic expressions with confidence and ease.