Problem 15 - Elementary Algebra

Question

For the following problems, simplify each of the square root expressions. $$ (4-\sqrt{5 y})^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(16 - 8\sqrt{5y} + 5y\).

Step 1: Identify the terms in the expression

In this problem, we have a binomial expression of the form \((4-\sqrt{5y})^2\). Here, our first term (a) is 4 and the second term (b) is \(\sqrt{5y}\).

Step 2: Apply the binomial square formula

Now, we will apply the formula (a-b)^2 = a^2 - 2ab + b^2 to our expression. In this case, a = 4 and b = \(\sqrt{5y}\). So the formula becomes: \((4 - \sqrt{5y})^2 = 4^2 - 2(4)(\sqrt{5y}) + (\sqrt{5y})^2\)

Step 3: Square and multiply terms

Next, we will square the terms and multiply the constants: \(16 - 8\sqrt{5y} + 5y\)

Step 4: Write the simplified expression

The simplified expression after all the calculations is: \((4-\sqrt{5y})^2 = 16 - 8\sqrt{5y} + 5y\)

Key Concepts

Binomial Expressions

A binomial expression is a type of algebraic expression that consists of exactly two distinct terms combined by either a plus or a minus sign. In our exercise, the expression \((4 - \sqrt{5y})^2\) is a classic example of a binomial expression. Such expressions are widely used in algebra and play a key role in polynomial operations. Understanding binomials is crucial because they form the basis for binomial expansion and simplification tasks.When dealing with binomial expressions, it’s important to identify the two parts, often labeled as \(a\) and \(b\). For instance, for the expression \((4 - \sqrt{5y})\), here, \(a\) is \(4\) and \(b\) is \(\sqrt{5y}\). Recognizing these individual components allows us to apply specific algebraic formulas, such as the Binomial Square Formula, to further simplify or expand the expression.

Algebraic Simplification

Algebraic simplification involves reducing expressions into their simplest forms while keeping their value equivalent. Simplifying often makes expressions easier to understand or work with, especially in solving equations or in advanced mathematical operations. In the context of this exercise, you are simplifying the expression \((4 - \sqrt{5y})^2\). This process includes using appropriate algebraic formulas and performing operations like squaring or multiplying terms.Why simplify?

• To make complex expressions manageable and easier to interpret.
• To facilitate further mathematical calculations like solving equations.
• To provide neat and compact results, useful in both academic and practical scenarios.

Simplification sometimes requires careful handling of square roots and other radicals, especially when part of a binomial expression. In this expression, we ended up with a simplified form of \(16 - 8\sqrt{5y} + 5y\), ready to be used for further calculations, if necessary.

Binomial Square Formula

The Binomial Square Formula is a powerful tool for expanding squared binomial expressions. It states that:\((a - b)^2 = a^2 - 2ab + b^2\) or, if the operation is addition, \((a + b)^2 = a^2 + 2ab + b^2\).In this formula, each term is treated separately:

• \(a^2\): the square of the first term.
• \(- 2ab\) (or \(+ 2ab\) for addition): twice the product of both terms.
• \(b^2\): the square of the second term.

For our specific problem, \((4 - \sqrt{5y})^2\), we used this formula to get:\[4^2 - 2\times4\times\sqrt{5y} + (\sqrt{5y})^2\], which simplifies to \(16 - 8\sqrt{5y} + 5y\).This formula is extremely useful, simplifying the process of handling powers of binomials without directly attempting to multiply them out by hand. It’s a shortcut grounded in algebraic identities that saves time and effort.