Problem 88 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \frac{(a b)^{3}}{a^{4} b^{-4}} $$

Step-by-Step Solution

Answer

Simplified expression is \(\frac{b^7}{a}\).

Step 1: Apply the Power of a Product Rule

The expression \((ab)^3\) means that both \(a\) and \(b\) are raised to the third power. This can be re-written using the power of a product rule: \((ab)^3 = a^3 b^3\). Substituting this into the expression gives: \(\frac{a^3 b^3}{a^4 b^{-4}}\).

Step 2: Apply Properties of Exponents to the Denominator

The denominator \(a^4 b^{-4}\) can be simplified by recognizing that a negative exponent indicates a reciprocal. Thus \(b^{-4} = \frac{1}{b^4}\). So, the expression simplifies to \(\frac{a^3 b^3}{a^4} \cdot b^4\).

Step 3: Simplify Using Division of Exponents

We will simplify \(\frac{a^3}{a^4}\) and \(\frac{b^3 \cdot b^4}{1}\). Using the property of exponents that states \(\frac{x^m}{x^n} = x^{m-n}\), we get \(a^{3-4} = a^{-1}\) and \(b^{3+4} = b^7\). So, we have \(a^{-1} b^7\).

Step 4: Express Using Positive Exponents

Finally, use the property that \(x^{-n} = \frac{1}{x^n}\) to convert the negative exponent to a positive. So, \(a^{-1} = \frac{1}{a}\). The expression becomes \(\frac{b^7}{a}\).

Key Concepts

Power of a Product Rule

When faced with an expression such as \((ab)^3\), you're dealing with the Power of a Product Rule. This rule states that when you raise a product to a power, you can independently apply that power to each factor of the product. Think of it as distributing the power to each element inside the parentheses. This means:

• \((ab)^3 = a^3 \cdot b^3\).

The idea behind this is very simple: You are multiplying the base (which is now two numbers multiplied together) by itself as many times as indicated by the exponent.
It helps break down more complex expressions into simpler, manageable factors.
So, whenever you see a product of numbers or variables in parentheses raised to a power, remember that this rule will come in handy.Use it to simplify expressions as you build towards the final desired form.

Negative Exponent

Negative exponents can be a little tricky at first, but they have a clear and useful rule.A negative exponent means you should take the reciprocal of the base and then apply the positive exponent.In simpler terms:

• \(x^{-n} = \frac{1}{x^n}\)

This rule helps in simplifying expressions and converting negative exponents to positive, which is often required.Let's use the denominator from the original exercise, \(b^{-4}\). By applying the rule for negative exponents, you convert it to:

• \(b^{-4} = \frac{1}{b^4}\)

This conversion not only simplifies expressions but also helps when combining other terms in divisions or products.
Over time, understanding and applying the rule for negative exponents will become second nature, allowing for more efficient problem resolution.

Division of Exponents

When dividing terms with the same base raised to exponents, there's a neat rule that makes simplification straightforward:

• \(\frac{x^m}{x^n} = x^{m-n}\)

This rule states that you subtract the exponent of the denominator from the exponent of the numerator.
For example, consider \(\frac{a^3}{a^4}\).Here, both 'a's are bases, and you're dividing the terms, leading to a subtraction of exponents:

• \(a^{3-4} = a^{-1}\)

Similarly, if you multiply terms with the same base, you add the exponents.In the original problem, combining \(b^3 \cdot b^4\) results in:

• \(b^{3+4} = b^7\)

These rules allow us to cleanly handle expressions and exponents, funneling them into simpler terms.Mastering these techniques is key to algebra efficiency, making complex divisions and multiplications easy to navigate.