Problem 70 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{\left(a^{6} b^{-2}\right)^{4}}{\left(4 a^{-3} b^{-3}\right)^{3}} $$

Step-by-Step Solution

Answer

\(\frac{a^{33} b}{64}\)

Step 1: Apply the Power of a Power Rule

First, consider the numerator \((a^6 b^{-2})^4\). Apply the power of a power rule, which states \((x^m)^n = x^{m \cdot n}\). This results in \(a^{6 \cdot 4} b^{-2 \cdot 4} = a^{24} b^{-8}\).

Step 2: Apply the Power of a Product Rule to the Denominator

For the denominator \((4a^{-3}b^{-3})^3\), first recognize that each term in the parentheses is raised to the power of 3. Apply the power of a product rule: \((xy)^n = x^n y^n\). This gives \(4^3 a^{-3 \cdot 3} b^{-3 \cdot 3}\), which simplifies to \(64 a^{-9} b^{-9}\).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

At this point, we have \(\frac{a^{24} b^{-8}}{64 a^{-9} b^{-9}}\). Use the property of exponents \(\frac{x^m}{x^n} = x^{m-n}\) separately for \(a\) and \(b\). The result is \(a^{24 - (-9)} b^{-8 - (-9)} = a^{33} b^1 = a^{33} b\). Note that the constant term remains in the denominator because there is no 64 in the numerator.

Step 4: Combine Results

Combine your results to express the original expression with positive exponents fully, ensuring that constants are handled properly. The final expression is \(\frac{a^{33} b}{64}\), which is in the simplified form with only positive exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra means making them easier to understand or work with. The goal is to reduce expressions to their simplest form, while still maintaining their original value. This often involves combining like terms, removing redundant elements, or using algebraic rules to condense expressions.

When simplifying expressions, it's important to:

• Apply relevant rules of arithmetic and algebra, such as distributing powers across terms.
• Re-organize terms to combine like elements or cancel opposite elements.
• Express the result with only positive exponents, which makes the expression cleaner and easier to interpret.

In the exercise we're looking at, simplifying means applying the power rules correctly, dividing the terms, and ensuring that all exponents are positive in the final result.

Power of a Power Rule

The power of a power rule is a fundamental concept in exponentiation, which tells us how to handle expressions where a power is raised to another power. Mathematically, it is expressed as \((x^m)^n = x^{m \cdot n}\). This means you multiply the exponents.

To apply the power of a power rule correctly:

• Identify the base and the exponents involved.
• Multiply the exponents together, keeping the base the same.
• Use this to simplify both the numerator and the denominator when necessary.

In our example, applying the power of a power rule helped to simplify both the numerator \((a^6 b^{-2})^4\) to \(a^{24} b^{-8}\) and the denominator \((4a^{-3}b^{-3})^3\) to \(64 a^{-9} b^{-9}\), making it possible to further simplify the overall expression.

Positive Exponents

Using positive exponents is often the preferred way to express algebraic expressions, as it simplifies understanding and computation. An exponent indicates how many times a number (the base) is multiplied by itself.

To convert negative exponents to positive exponents:

• Remember the rule: \(x^{-n} = \frac{1}{x^n}\), which means that negative exponents "invert" the base.
• Ensure all terms in the final expression have positive exponents for clarity and simplicity.
• Simplify fractions by applying the rule \(\frac{x^m}{x^n} = x^{m-n}\), being careful with the signs of the exponents.

In our solution, we took care to convert all the exponents to positive by rearranging the terms, for example, \(b^{-8 - (-9)}\) became \(b^1\), which is simply \(b\). This conversion allowed us to express the result purely in terms of positive exponents, making it look neater as \(\frac{a^{33} b}{64}\).