Problem 99 - Algebra A Combined Function

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

The expression in the denominator is \((x^5 y^2)^{-2}\). To simplify this, we use the property \((x^m)^{-n} = (x^{-m})^{n} = \frac{1}{(x^m)^n}\). Therefore, \((x^5 y^2)^{-2} = \frac{1}{(x^5 y^2)^2}\). Simplifying further, this becomes \(x^{-10} y^{-4}\). Now, the overall expression becomes: \[\frac{7 x^{-1} y}{14 x^{-10} y^{-4}}\] .

Step 2: Simplify the Coefficients and Cancel Common Factors

In the expression \(\frac{7}{14}\), the coefficients can be simplified: \(\frac{7}{14} = \frac{1}{2}\). Now, combine all parts: \[\frac{1}{2} \cdot \frac{x^{-1} y}{x^{-10} y^{-4}}\].

Step 3: Simplify the Variables Using Exponent Rules

Apply the quotient rule for exponents, \(\frac{x^a}{x^b} = x^{a-b}\), to both \(x\) and \(y\). For \(x\): \(x^{-1} - (-10) = x^{9}\) and for \(y\): \(y^{1} - (-4) = y^{5}\). Thus, the expression simplifies to \[ \frac{1}{2} x^9 y^5.\]

Key Concepts

Simplifying Expressions

When working with algebraic expressions, simplifying expressions is a fundamental skill. The goal is to write the expression in its simplest form to make calculations easier. Simplifying involves combining like terms, factoring, and using the rules of arithmetic and exponents.
In the given exercise, we are tasked with simplifying an expression that includes both coefficients and variables with exponents. By breaking down each component—such as the numerator and the denominator—we can apply specific algebraic methods to simplify the entire expression. Think of it as decluttering your workspace; the fewer unnecessary parts, the cleaner and more efficient it becomes.

Exponent Rules

Exponent rules are essential when dealing with expressions containing powers or exponents. These rules help in manipulating and simplifying expressions easily.

• Product Rule: If you multiply like bases, you add their exponents, \((x^a \cdot x^b = x^{a+b})\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents, \((x^{a})^{b} = x^{a\cdot b}\).
• Negative Exponent Rule: Negative exponents denote reciprocals, \(x^{-a} = \frac{1}{x^a}\).
• Zero Exponent Rule: Any base (except zero) raised to the power of zero is one, \(x^0 = 1\).

The proper use of these rules allows us to transform complex expressions into simpler forms, as shown in the first step of the solution where the denominator was simplified using these rules.

Quotient Rule for Exponents

The quotient rule for exponents states that when you divide two numbers of the same base, you subtract the exponents. This rule is expressed as \(\frac{x^a}{x^b} = x^{a-b}\).
This rule is incredibly useful when simplifying fractional expressions involving variables with exponents. In the exercise, we applied the quotient rule to the variables \(x\) and \(y\) separately.
By subtracting the exponents of the corresponding bases in the numerator and the denominator, we simplified the expression significantly. For instance, \(x^{-1} - (-10)\) results in \(x^9\), and \(y^{1} - (-4)\) gives \(y^5\). This crucial step shows how the quotient rule helps turn a rather intricate expression into something more manageable and elegant.