Problem 92 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{33 a^{-4} b^{-7}}{11 a^{3} b^{-2}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{33a^{-4} b^{-7}}{11a^3 b^{-2}}\) and rewrite it using only positive exponents. Answer: \(\frac{3}{a^7 b^5}\)

Step 1: Simplify the coefficients

First, we will divide the constant coefficients: $$\frac{33}{11} = 3$$. Our expression now becomes: $$ \frac{3 a^{-4} b^{-7}}{a^{3} b^{-2}} $$.

Step 2: Use properties of exponents

We will use the properties of exponents: - \(\frac{a^x}{a^y} = a^{x-y}\) - \(a^{-x} = \frac{1}{a^x}\) Applying these properties to the expression, we get: $$ \frac{3 a^{-4} b^{-7}}{a^{3} b^{-2}} = 3a^{-4-3}b^{-7-(-2)} = 3a^{-7}b^{-5} $$.

Step 3: Convert the negative exponents to positive exponents

Now to change the negative exponents to positive, we will move them accordingly from numerator to denominator or vice versa. We have: $$ 3a^{-7}b^{-5}= 3\frac{1}{a^7}\frac{1}{b^5} = \frac{3}{a^7 b^5} $$. The final expression with only positive exponents: $$ \frac{3}{a^7 b^5} $$.

Key Concepts

Understanding the Properties of Exponents

When dealing with exponents, it's essential to know some key properties that make simplifying and solving exponent-related expressions straightforward. These properties help manipulate expressions elegantly, without altering their values. Let's explore the two primary exponent rules utilized within the example provided:

• Power of a Quotient: The rule \(\frac{a^x}{a^y} = a^{x-y}\) allows us to divide two numbers with the same base by subtracting the exponent in the denominator from the exponent in the numerator.
• Negative Exponent Rule: This rule states that \(a^{-x} = \frac{1}{a^x}\), meaning a negative exponent indicates the reciprocal of the base raised to a positive exponent.

These properties are foundational in operations involving exponents and allow the conversion of complex expressions into simpler forms, which is especially useful when working with both positive and negative exponents.

Simplifying Expressions Using Exponents

To simplify expressions involving exponents, understanding and applying exponent properties are crucial. Let's consider the given expression, breaking it down step by step.

Initially, the expression \(\frac{33 a^{-4} b^{-7}}{11 a^{3} b^{-2}}\) is simplified by directing focus towards the coefficients first. Dividing the coefficients gives \(\frac{33}{11} = 3\). This simplification reduces complexity early on, transforming the expression to \(\frac{3 a^{-4} b^{-7}}{a^{3} b^{-2}}\).

Next, the properties of exponents are applied to simplify the variables. By subtracting the exponents of like bases \(a\) and \(b\) found in the numerator and the denominator (e.g., \(a^{-4-3}\) and \(b^{-7-(-2)}\)), the expression simplifies to \(3a^{-7}b^{-5}\). This approach smartly consolidates the terms into a manageable form, facilitating the final steps for simplification.

Handling Negative Exponents

Negative exponents can initially seem intimidating, but they merely indicate a reciprocal relationship. In mathematical expressions, transforming negative exponents into positive ones makes the expression easier to understand and work with.

In the given exercise, the conversion of \(3a^{-7}b^{-5}\) to an expression with positive exponents is accomplished by recognizing \(a^{-7}\) and \(b^{-5}\) can respectively be rewritten as \(\frac{1}{a^7}\) and \(\frac{1}{b^5}\). This transformation is based on the negative exponent rule, which switches the positions of terms from the numerator to the denominator or vice versa, thus flipping their exponent sign.

Therefore, \(3a^{-7}b^{-5}\) becomes \(\frac{3}{a^7 b^5}\), a neat expression with all positive exponents. This not only simplifies calculations but also aligns with convention, making it intuitive for further mathematical operations or interpretations.