Problem 41 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(\frac{7 a^{4}}{b^{-1}}\right)^{-2}$$

Step-by-Step Solution

Answer

The short answer is: \(\frac{b^2}{49a^8}\)

Step 1: Distribute the exponent to each term inside the parentheses

Apply the power rule \((a^m)^n = a^{mn}\) to distribute the exponent -2 to each term inside the parentheses: \(7^{-2}a^{4(-2)}b^{(-1)(-2)}\)

Step 2: Simplify the expression further by applying exponent rules

Now, simplify each term using the exponent rules, \(7^{-2} = \frac{1}{7^2}, a^{4(-2)} = a^{-8}, b^{(-1)(-2)} = b^2\) So the expression becomes: \(\frac{1}{7^2}a^{-8}b^2\)

Step 3: Eliminate the negative exponent

Since we are asked to find the expression without negative exponents, we can use the rule \(a^{-n} = \frac{1}{a^n}\) for the term \(a^{-8}\), which becomes: \(\frac{1}{a^8}\).

Step 4: Combine the terms

Now, combine all the terms together: \[ \frac{1}{7^2}\times\frac{1}{a^8}\times b^2 = \frac{b^2}{49a^8} \] So the simplified expression is: \[ \frac{b^2}{49a^8} \]

Key Concepts

Understanding Negative Exponents

Negative exponents can be tricky, but they're not as complicated as they may sound. Essentially, a negative exponent signifies that the base is on the opposite side of the fraction line compared to where we see it. For example, when you see a negative exponent like

• \(a^{-n}\), it means the same as \(\frac{1}{a^n}\).

Think of it as a way to "flip" the base across the fraction bar. This flip is crucial, especially when simplifying expressions.Using this concept, we can confidently tackle expressions like \(a^{-8}\). The negative sign tells us to flip the base. Thus, \(a^{-8}\) becomes \(\frac{1}{a^8}\). This adjustment helps create clarity in the expression without any negativity involved!

Applying the Power Rule

The power rule is a useful tool in algebra that aids in simplifying expressions with exponents. The rule states that when raising a power to another power, you multiply the exponents. Formally, it's written as \((a^m)^n = a^{mn}\).This means that you take the base, multiply the exponents, and keep the base the same.For example, in our exercise, the term \(a^{4(-2)}\) becomes \(a^{-8}\) by multiplying the 4 by -2.Applying the power rule can simplify complex expressions quickly by reducing multiple exponent operations to a single step.This rule applies equally to numerical and algebraic bases, ensuring that the approach is consistent and straightforward across different parts of an expression.Remember, practice makes perfect with the power rule. The more you use it, the more natural it will become.

Mastering Simplifying Expressions

Simplifying expressions is about making them as straightforward as possible, minimizing complexity without changing their value. This involves

• replacing negative exponents with their positive counterparts
• combining like terms
• applying various algebraic rules.

Simplification usually starts by addressing any negative exponents and converting them to positive forms. This ensures no negativity in the final expression - for instance, switching \(a^{-8}\) to \(\frac{1}{a^8}\).Next, each part of the expression is combined in a way that resultsin the simplest form possible. In our example, this meant rewriting and combining fractions to give \(\frac{b^2}{49a^8}\).This process of simplification helps to make the function easier to understand and to work with in further mathematical operations, facilitating deeper understanding and analysis.