Problem 82 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \(\left(x^{1 / 2}+y^{1 / 2}\right)\left(x^{1 / 2}-y^{1 / 2}\right)\)

Step-by-Step Solution

Answer

The simplified form is \(x - y\).

Step 1: Recognize the Expression Type

The expression given is in the form \((a+b)(a-b)\) which is a difference of squares.

Step 2: Apply the Difference of Squares Formula

Recognizing the form, we can use the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\). Here \(a = x^{1/2}\) and \(b = y^{1/2}\).

Step 3: Square the Terms

Calculate \(a^2\) and \(b^2\). For \(a = x^{1/2}\), \(a^2 = (x^{1/2})^2 = x\). Similarly, for \(b = y^{1/2}\), \(b^2 = (y^{1/2})^2 = y\).

Step 4: Simplify the Expression

Substitute back in the formula to find the simplified form: \(a^2 - b^2 = x - y\). Thus, \((x^{1/2} + y^{1/2})(x^{1/2} - y^{1/2}) = x - y\).

Key Concepts

Difference of Squares

The difference of squares is a unique pattern in algebra that can greatly simplify expressions. It involves two binomials that have a specific form:

• The first binomial is the sum of two terms
• The second is the difference of those same terms.

Mathematically, it's represented as \[(a+b)(a-b) = a^2 - b^2\].

This is known as the difference of squares formula and is particularly handy because it simplifies calculations and expressions. By recognizing patterns such as this, you can break down complex problems into simpler parts. For example, in expressions like \((x^{1/2}+y^{1/2})(x^{1/2}-y^{1/2})\), recognizing that it fits the \((a+b)(a-b)\) pattern allows you to directly simplify it to \(x - y\) by squaring the individual components.

Simplifying Expressions

Simplifying expressions in algebra means making them as concise as possible while keeping their values unchanged. It's like cleaning up clutter in a room so that you can better understand what's there.

The process usually involves:

• Combining like terms
• Using mathematical identities
• Applying algebraic rules to transform the expression into its simplest form

When provided with an algebraic expression like \((x^{1/2}+y^{1/2})(x^{1/2}-y^{1/2})\), recognizing it as a difference of squares helps you proceed directly to its simplified form \(x-y\).

It saves time and reduces the chance for error compared to multiplying the terms individually and then simplifying.

Exponent Rules

Exponent rules help manage the powers of numbers and variables in algebraic expressions. Understanding these rules allows for easier manipulation of expressions with exponents.

There are several key exponent rules:

• When multiplying like bases, add the exponents: \(a^m \times a^n = a^{m+n}\)
• When dividing like bases, subtract the exponents: \(a^m / a^n = a^{m-n}\)
• A power of a power means you multiply the exponents: \((a^m)^n = a^{m \times n}\)
• A number raised to the zero power is always 1: \(a^0 = 1\) if \(aeq 0\)

For the expression \((x^{1/2}+y^{1/2})(x^{1/2}-y^{1/2})\), applying the exponent rule for squaring \(x^{1/2}\) results in \((x^{1/2})^2 = x^{1}\), simplifying it to \(x\). Similarly, \((y^{1/2})^2 = y\). These exponent rules offer a way to simplify expressions without needing detailed calculations at every step.