Problem 125 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression.$$x y\left(x^{-1}+y^{-1}\right)^{-1}$$

Step-by-Step Solution

Answer

The simplified expression is \(2xy\).

Step 1: Distribute the Outer Expression

Distribute the \(xy\) term inside the bracket to each term.\(xy(x^{-1}+y^{-1})\) becomes \(x^{2}y \cdot x^{-1} + xy^{2} \cdot y^{-1}\)

Step 2: Simplify Using Exponent Rules

The rule for multiplications with same base is: \(a^{n} \cdot a^{m} = a^{n+m}\). Hence, simplify the expressions in Step 1 to get: \(x^{2-1}y + xy^{2-1}\) which results in \(xy + xy\)

Step 3: Combine Like Terms

Add the two similar terms obtained in step 2. This gives \(2xy\)

Key Concepts

Exponent Rules

Exponent rules are essential when simplifying algebraic expressions, as they determine how to handle powers of numbers or variables. One of the most fundamental rules is the product of powers rule: \(a^{n} \cdot a^{m} = a^{n+m}\). This rule tells us that when we multiply two powers with the same base, we add the exponents.

In our exercise, applying this rule is crucial. When distributing \(xy\) into \(x^{-1}+y^{-1}\), we look at terms like \(x^{2}y \cdot x^{-1}\). Using the product of powers rule, we simplify \(x^{2-1}\), resulting in \(x^{1}\), which is simply \(x\).

Likewise, for the term \(xy^{2} \cdot y^{-1}\), using the same rule, \(y^{2-1}\) becomes \(y\). Understanding these exponent rules allows us to simplify expressions correctly and is vital in algebra.

Distribution in Algebra

The distributive property in algebra is a critical concept that helps us understand how to multiply across terms within parentheses. It allows us to simplify expressions by ensuring each term in the parenthesis is multiplied by the factor outside.

In this example, the expression \(xy(x^{-1}+y^{-1})\) uses distribution to multiply \(xy\) with both \(x^{-1}\) and \(y^{-1}\). This step results in two separate terms: \(x^{2}y \cdot x^{-1}\) and \(xy^{2} \cdot y^{-1}\).

• First, multiply \(xy\) with \(x^{-1}\) to get \(x^{2}y \cdot x^{-1}\).
• Second, multiply \(xy\) with \(y^{-1}\) to obtain \(xy^{2} \cdot y^{-1}\).

This distribution step helps break down complex expressions into simpler parts, making them more manageable to solve.

Combining Like Terms

Combining like terms is a straightforward but powerful tool that can significantly simplify algebraic expressions. Like terms are terms that share the same variable factors raised to the same power. Once identified, we can easily add or subtract these terms.

After using distribution and applying exponent rules in our exercise, we moved to combine like terms. We ended with \(xy + xy\). Notice that these terms are alike because both have the variables \(x\) and \(y\) raised to the same power. Therefore, they can be combined.

• Combine terms \(xy + xy\) to simplify the expression to \(2xy\).

By recognizing and combining like terms, we streamline our algebraic expression to its simplest form, giving us a clear and concise result.