Problem 32 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{100 y}-\sqrt{25 y} $$

Step-by-Step Solution

Answer

The expression simplifies to \(5\sqrt{y}\).

Step 1: Simplify Each Square Root

First, simplify each square root separately. The term \(\sqrt{100y}\) can be rewritten by recognizing that 100 is a perfect square: \(\sqrt{100y} = \sqrt{100} \cdot \sqrt{y} = 10\sqrt{y}\). Similarly, for \(\sqrt{25y}\), note that 25 is also a perfect square: \(\sqrt{25y} = \sqrt{25} \cdot \sqrt{y} = 5\sqrt{y}\).

Step 2: Subtract the Simplified Expressions

Now subtract the simplified expressions obtained in Step 1. Perform the subtraction: \(10\sqrt{y} - 5\sqrt{y}\). Since both terms have \(\sqrt{y}\) as a common factor, you can factor it out: \((10 - 5)\sqrt{y} = 5\sqrt{y}\).

Step 3: Verify the Expression Is in Simplest Form

Check to ensure that the resulting expression \(5\sqrt{y}\) is in its simplest form. There are no common factors left to factor further, and \(\sqrt{y}\) is already simplified. Thus, \(5\sqrt{y}\) is the simplest form of the expression.

Key Concepts

Perfect Squares

Perfect squares are numbers that are the square of an integer. In other words, when you multiply an integer by itself, you get a perfect square. For example, 100 is a perfect square because it is the result of multiplying 10 by itself: \(10 \times 10 = 100\). Perfect squares are useful when simplifying square roots because these numbers allow radicals to be expressed as whole numbers.

• For example, \( \sqrt{100} = 10 \) because 100 is a perfect square.
• Similarly, \( \sqrt{25} = 5 \) because 25 is a perfect square.

Recognizing perfect squares helps to quickly simplify radical expressions since finding the square root of a perfect square yields an integer.

Factoring

Factoring is the process of breaking down an expression into products of other expressions that are multiplied together. In the context of the step-by-step solution, factoring was used when subtracting the expressions. After simplifying, both terms share a common factor, which is \( \sqrt{y} \).

• For instance, in the expression \(10\sqrt{y} - 5\sqrt{y}\), \( \sqrt{y} \) is a common factor that can be factored out.
• This allows the expression to be written as \((10 - 5)\sqrt{y} = 5\sqrt{y}\).

Factoring helps in simplifying the expression further by revealing any commonalities in the terms that can be condensed.

Square Roots

A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because \(5 \times 5 = 25\). When working with variables, the concept remains fundamentally the same. Square roots are often used in algebraic expressions, especially when simplifying radicals.

• In the exercise, \( \sqrt{100y} \) was separated into \( \sqrt{100} \) and \( \sqrt{y} \), which simplifies to \( 10\sqrt{y} \).
• Similarly, \( \sqrt{25y} \) simplifies to \( 5\sqrt{y} \) by taking the square root of 25 and leaving \( \sqrt{y} \) in place.

Understanding square roots allows you to break down and simplify radical expressions efficiently.

Simplifying Expressions

Simplifying expressions involves reducing expressions to their simplest form. This process often includes factoring, recognizing and using perfect squares, and simplifying square roots. Once expressions are simplified, they are easier to work with and understand.
To simplify the expression in the exercise:

• First, identify and simplify the square roots by recognizing perfect squares.
• Next, factor out any common terms like \( \sqrt{y} \) from the expressions.
• Finally, perform any basic arithmetic actions like addition or subtraction to finalize the simplest form.

This process allows us to take a potentially complex expression and make it much easier to handle, as seen in the example where \(10\sqrt{y} - 5\sqrt{y}\) simplifies to \(5\sqrt{y}\).