Problem 43 - 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. $$\frac{12^{5 / 4} y^{-2}}{12^{-1} y^{-3}}$$

Step-by-Step Solution

Answer

The result is \(12^{9/4} y\).

Step 1: Simplify the Numerator and Denominator

We begin by simplifying the given expression: \( \frac{12^{5 / 4} y^{-2}}{12^{-1} y^{-3}} \). The expression in the numerator is \( 12^{5/4} y^{-2} \) and the expression in the denominator is \( 12^{-1} y^{-3} \).

Step 2: Apply the Quotient Rule

Now, apply the quotient rule which states \( \frac{a^m}{a^n} = a^{m-n} \) to both the base of 12 and the base of \( y \). This gives us:\[\frac{12^{5/4}}{12^{-1}} = 12^{5/4 - (-1)} = 12^{5/4 + 1}\]\[\frac{y^{-2}}{y^{-3}} = y^{-2 - (-3)} = y^{-2 + 3}\]

Step 3: Calculate the Exponents

Simplify the exponents derived from the quotient rule:- For the base of 12: - \[ 12^{5/4 + 1} = 12^{5/4 + 4/4} = 12^{9/4} \]- For the base of \( y \): - \( y^{-2 + 3} = y^{1} = y \)

Step 4: Final Expression with Positive Exponents

Combine the simplified expressions with positive exponents:- The final expression is: \[12^{9/4} y\]

Key Concepts

Quotient Rule

The quotient rule is an essential concept in exponentiation that helps simplify expressions involving division of like bases. When you have a term in the form of \( \frac{a^m}{a^n} \), the quotient rule tells us that we can write this as \( a^{m-n} \). This rule works because you are effectively canceling out the common factors in both the numerator and the denominator.

Steps to Apply the Quotient Rule:

• Observe the powers of the like bases (here, \(12\) and \(y\)).
• Subtract the exponent in the denominator from the exponent in the numerator.
• Re-write the expression using the new exponent.

In our exercise, applying the quotient rule to the base 12, we calculate \( \frac{12^{5/4}}{12^{-1}} = 12^{5/4 - (-1)} = 12^{9/4} \). Similarly, for \(y\), we find \( \frac{y^{-2}}{y^{-3}} = y^{-2 - (-3)} = y^1 \).

Using the quotient rule simplifies the process, making it easier to handle complex expressions quickly.

Positive Exponents

Working with positive exponents is crucial because it keeps expressions straightforward and eliminates confusion related to negative signs. A positive exponent indicates how many times a number, known as the base, is multiplied by itself. Negative exponents, on the other hand, indicate the reciprocal of the base raised to the corresponding positive power.

Steps to Ensure Positive Exponents:

• Convert any negative exponent \(-a\) into \( \frac{1}{base^a} \).
• Add any necessary number to turn the exponent positive, often using algebraic manipulation.
• Rewrite the expression with positive exponents.

In the exercise, the exponent for \(y\) was simplified to a positive exponent by recognizing that \(y^{-2 + 3} = y^1\). For the base of 12, we performed \(5/4 + 1\) to achieve \(9/4\), leading to positive exponents throughout. Maintaining positive exponents in solutions is key for clarity and ensuring that expressions are universally understood.

Simplifying Expressions

Simplifying expressions involves rewriting them in their most basic or understandable form without changing their value. This process often requires applying several algebraic rules and can include operations such as combining like terms, factoring, and using exponent laws.

Steps for Simplifying:

• Apply any rules of exponents, like the quotient rule, to reduce the number of terms.
• Combine like terms if they exist.
• Ensure all exponents are positive and simplify any fractions.

In our problem, simplifying the expression from \( \frac{12^{5/4} y^{-2}}{12^{-1} y^{-3}} \) to \( 12^{9/4} y \) involves applying the quotient rule, ensuring positive exponents, and reducing the terms to their simplest form. This step-by-step simplification is central to mastering algebraic expressions, making them easier to interpret and utilize.