Problem 97 - Intermediate Algebra

Question

Simplify each exponential expression. $$ \frac{7 x^{-1} y}{14\left(x^{5} y^{2}\right)^{-2}} $$

Step-by-Step Solution

Answer

\(\frac{1}{2} x^9 y^5\)

Step 1: Simplify the Denominator Exponentially

In the denominator, we have \((x^5 y^2)^{-2}\). Apply the power of a power property, which states \((a^m)^n = a^{m imes n}\). So \((x^5 y^2)^{-2} = x^{-10} y^{-4}\). This means the denominator can be rewritten as \(x^{-10} y^{-4}\).

Step 2: Rewrite the Entire Expression

Substitute the rewritten denominator into the expression. We have: \[ \frac{7 x^{-1} y}{14 x^{-10} y^{-4}} \] This means we'll simplify the fraction by dealing with each variable separately.

Step 3: Simplify the Coefficients

The expression can be simplified by dividing the numerical coefficients: \(\frac{7}{14} = \frac{1}{2}\). So the coefficient in front of the expression becomes \(\frac{1}{2}\).

Step 4: Apply Quotient Rule to Exponents

Apply the quotient of powers property \( \frac{a^m}{a^n} = a^{m-n} \) for each variable. This transforms our expression: 1. For \(x\), \(x^{-1} - (-10) = x^{9}\) 2. For \(y\), \(y^1 - (-4) = y^{5}\).Thus, the simplified expression is \(\frac{1}{2} x^{9} y^{5}\).

Key Concepts

Simplifying Fractions

To simplify fractions, whether they involve numbers, variables, or exponential expressions, the key is to reduce them into their simplest form. In the given problem, the fraction consists of exponential terms in both the numerator and the denominator. Simplifying involves reducing these terms step by step.

Here's a simple approach to simplifying fractions:

• Begin by dividing the numerical coefficients, like how we divided 7 by 14, resulting in \(\frac{1}{2}\).
• Then, process any like bases with exponents using properties like the quotient of powers.
• Adjust the exponents accordingly so that the entire expression becomes easier to handle.

Breaking fractions down helps reveal the underlying structure, making it easier to apply mathematical rules and simplify them effectively.

Quotient of Powers

The quotient of powers property is a handy tool when you're simplifying expressions containing like bases raised to exponents. This principle can be understood through the formula \(\frac{a^m}{a^n} = a^{m-n}\), which means if you have the same base, you can subtract the exponent in the denominator from the exponent in the numerator.

In the problem we're considering:

• For the base \(x\), \(x^{-1} - (-10)\) became \(x^{9}\).
• For the base \(y\), we computed \(y^{1} - (-4)\) to get \(y^{5}\).

It simplifies the expression by combining terms that share the same base, streamlining complex exponential expressions. This tool is essential when working with multiple variables and exponents in calculations.

Negative Exponents

Negative exponents can seem tricky at first, but they are actually straightforward once you get the hang of them. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In mathematics, it's expressed as \(a^{-n} = \frac{1}{a^n}\).

In our example, we faced terms with negative exponents:

• In the denominator: \((x^5 y^2)^{-2}\) was rewritten to \(x^{-10} y^{-4}\).
• Using the reciprocal concept, these terms reflect flipping the base to the opposite position in the fraction.

Understanding negative exponents is crucial for simplifying expressions without extra steps, and making the overall expression more manageable by clarifying its position in a fraction.

Power of a Power Property

The power of a power property is an essential rule when working with exponential expressions. This property states that when raising a power to another power, you multiply the exponents together, expressed as \((a^m)^n = a^{m \times n}\).

This property was pivotal in our original exercise. For the denominator \((x^5 y^2)^{-2}\), it helped transform the expression into something more workable, simplifying each part to \(x^{-10} y^{-4}\).

When combinations of powers arise, this property simplifies the mathematical process, allowing the redistribution of factors without advancing into cumbersome calculations. It's a rule that stitches together multiple exponent manipulations into a neat solution.