Problem 61 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \left(\frac{a^{-5} b}{a b^{3}}\right)^{-4} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^{24} \cdot b^{8} \).

Step 1: Apply the Negative Exponent Rule

Start by recognizing that in the expression \( \left( \frac{a^{-5} b}{a b^{3}} \right)^{-4} \), there is a negative exponent \(-4\). Using the negative exponent rule \( x^{-n} = \frac{1}{x^n} \), we rewrite the expression as: \[ \left( \frac{a b^{3}}{a^{-5} b} \right)^{4} \]

Step 2: Simplify the Numerator and Denominator

Look at the fraction inside the parentheses: Numerator: \( a \cdot b^3 \) Denominator: \( a^{-5} \cdot b \)Simplify these by canceling common terms:The \( a \)'s can be simplified as \( a^{1} / a^{-5} = a^{1+5} = a^{6} \). For the \( b \)'s, simplify \( b^{3} / b^{1} = b^{3-1} = b^{2} \).The expression is now: \[ (a^{6} b^{2})^{4} \]

Step 3: Distribute the Exponent

Now distribute the exponent \(4\) to each factor inside the parentheses:Apply the power rule \((x^m)^n = x^{m \, * \, n}\) to both \(a^6\) and \(b^2\):\[ a^{6 \times 4} \cdot b^{2 \times 4} = a^{24} \cdot b^{8} \]

Step 4: Write the Answer Using Positive Exponents Only

The expression \( (a^{6} b^{2})^{4} \) simplifies to \( a^{24} \cdot b^{8} \), and since both exponents are positive, the final answer is:\[ a^{24} \cdot b^{8} \]

Key Concepts

Understanding Negative Exponents

Negative exponents might look a bit intimidating at first, but they are actually quite simple once you get the hang of them. The core idea is that a negative exponent indicates the reciprocal of that base raised to the corresponding positive exponent.
For example, if you see something like \( x^{-n} \), this simply means \( \frac{1}{x^n} \).
When working with negative exponents, remember to:

• Identify the negative exponent in the expression.
• Rewrite it as a reciprocal with a positive exponent.
• Simplify the expression further if possible.

The trickiest part is remembering that the negative sign doesn't mean the number is negative, just that we're dealing with a reciprocal. In our original exercise, we had \( \left(\frac{a^{-5} b}{a b^{3}}\right)^{-4} \), which turns into \( \left(\frac{a b^{3}}{a^{-5} b} \right)^{4} \) by using this very rule.

Simplifying Expressions with Exponents

Part of mastering algebra involves simplifying expressions, especially those that include exponents. Simplifying means combining like terms and canceling what you can. Here’s how it applies to our example:

Our task started with \( \frac{a^{-5} b}{a b^{3}} \). To simplify this, we tackled it by

• Identifying the like terms in the numerator and the denominator.
• You'll note that the \( a \'s \) are paired: \( a^{-5} \) and \( a \). By using exponent rules, we turn this into \( a^{-5-1} = a^1 \).
• For \( b \'s \), \( b \) and \( b^3 \) simplify to \( b^{3-1} = b^2 \).

This simplification step changes our original inner fraction to \( a^6 b^2 \), streamlining the overall expression.

Applying the Power Rule

The power rule is a fundamental part of working with exponents. It states that when you raise an exponential expression to another power, you multiply the exponents. In our case, we were given the expression \( (a^6 b^2)^4 \).

To apply the power rule here:

• Apply the power to each base individually.
• Multiply the original exponent by the new power: \( a^{6 \times 4} \) becomes \( a^{24} \) and \( b^{2 \times 4} \) becomes \( b^8 \).

The power rule simply requires recognizing the multiplication of exponents within parenthesis and executing correctly to get \( a^{24} \cdot b^{8} \), our simplified expression, now written with positive exponents only.