Problem 15 - Contemporary Precalculus

Question

Compute and simplify. $$(x+y)^{1 / 2}\left[(x+y)^{1 / 2}-(x+y)\right]$$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $(x+y)^{1/2}[(x+y)^{1/2}-(x+y)]$ Answer: $(x+y)^{1/4} - (x+y)^{3/2}$

Step 1: Identify the common expression

The expression \((x+y)^{1/2}\) is common to both terms inside the brackets.

Step 2: Apply the distributive property

Distribute the common expression \((x+y)^{1/2}\) to both terms inside the brackets: $$(x+y)^{1/2} \cdot (x+y)^{1/2} - (x+y)^{1/2} \cdot (x+y)$$

Step 3: Simplify the first term

Using the exponent rule for the product of like terms: \((a^m)^n = a^{mn}\), we have: $$(x+y)^{(1/2)(1/2)} - (x+y)^{1/2} \cdot (x+y)$$ Since \((1/2)(1/2) = 1/4\), the simplified expression becomes: $$(x+y)^{1/4} - (x+y)^{1/2} \cdot (x+y)$$

Step 4: Simplify the second term

Using the exponent rule for the product of like terms, we get: $$(x+y)^{1/4} - (x+y)^{(1/2)+1}$$ Since \((1/2)+1 = 3/2\), the simplified expression becomes: $$(x+y)^{1/4} - (x+y)^{3/2}$$ The final simplified expression is: $$\boxed{(x+y)^{1/4} - (x+y)^{3/2}}$$

Key Concepts

Distributive Property

Understanding the distributive property is essential for simplifying algebraic expressions. It allows you to multiply a single term by each term within a parenthesis in an expression. This principle can be written as: \( a(b + c) = ab + ac \). When you see an expression where a term is outside of a parenthesis, as in our exercise, you apply this property to 'distribute' the outside term to each of the terms inside.

For example, if you start with \( (x+y)^{1/2} \), and you need to distribute it over \( (x+y)^{1/2} - (x+y) \), you individually multiply \( (x+y)^{1/2} \) by each term in the brackets, which gives you two separate products. As demonstrated in the exercise, the distributive property simplifies the complex expression into more manageable parts that can be dealt with using other algebraic rules.

Exponent Rules

Exponent rules, or laws of exponents, are a set of rules that describe how to handle mathematical operations involving exponents. These rules are vital when dealing with expressions like the one provided. The most relevant rules for our exercise include:

• The Product of Powers Rule: \( a^m \cdot a^n = a^{m+n} \), which is used when you multiply terms with the same base.
• The Power of a Power Rule: \( (a^m)^n = a^{mn} \), which is used when an exponent is raised to another exponent.

Applying these rules allows us to simplify the terms \( (x+y)^{1/2} \cdot (x+y)^{1/2} \) and \( (x+y)^{1/2} \cdot (x+y) \) in the exercise. The first term represents the Power of a Power Rule and the second term illustrates the Product of Powers Rule. By manipulating the exponents according to these laws, you can combine and reduce the terms to reach a simpler expression.

Simplifying Radicals

Simplifying radicals involves expressing the radical—also known as the root of a number—in its simplest form. A radical is simplified when there are no perfect powers of the radicand (the number under the radical sign) remaining. When you work with expressions that contain roots, like \( (x+y)^{1/2} \), you are dealing with radicals.

In our exercise, we see a radical expression being raised to a power, such as \( (x+y)^{1/4} \). Simplifying such expressions might involve rationalizing the denominator or combining like terms with roots. Simplifying radicals is similar to simplifying exponents because radical expressions can also be expressed with rational exponents. This makes exponent rules applicable, providing a powerful tool for simplification. In general, approaching these problems requires understanding the relationship between exponents and roots, which ultimately leads to expressing the quantity in a more straightforward manner.