Problem 34 - 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. $$ \sqrt{18 b^{2}}+\sqrt{800 b^{2}} $$

Step-by-Step Solution

Answer

The simplest form is \( 23b\sqrt{2} \).

Step 1: Simplifying Each Square Root

We start by simplifying each square root separately. For \( \sqrt{18b^2} \), we factor inside the square root to get \( \sqrt{18b^2} = \sqrt{9 \times 2 \times b^2} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{b^2} \). Since \( \sqrt{9} = 3 \) and \( \sqrt{b^2} = b \), we have \( 3b\sqrt{2} \). For \( \sqrt{800b^2} \), we factor to get \( \sqrt{800b^2} = \sqrt{400 \times 2 \times b^2} = \sqrt{400} \cdot \sqrt{2} \cdot \sqrt{b^2} \). Since \( \sqrt{400} = 20 \) and \( \sqrt{b^2} = b \), we have \( 20b\sqrt{2} \).

Step 2: Adding the Simplified Expressions

Now, we add the two expressions we simplified: \( 3b\sqrt{2} + 20b\sqrt{2} \). Since both terms have the common factor \( b\sqrt{2} \), we combine the coefficients: \( (3 + 20)b\sqrt{2} = 23b\sqrt{2} \). This is our simplest form of the expression.

Key Concepts

Square Root Simplification

Simplifying square roots is an important skill in basic algebra, and it helps in reducing mathematical expressions to their simplest form.
Square root simplification involves breaking down the number inside the square root (radicand) into its prime factors and simplifying any perfect squares. To start with an example, let's consider the number 18. We can factor it as 9 multiplied by 2, because 9 is a perfect square. Thus, \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\).
Similarly, when dealing with variables, such as \(b^2\), if the index of the root is 2, the square root is simply \(b\), since \(b\cdot b = b^2\).

• Identify perfect square factors.
• Separate and simplify each perfect square.
• Multiply what’s left under the radical.

By applying these steps, you simplify square roots and make expressions more manageable.

Combining Like Terms

Combining like terms is an essential concept that simplifies algebraic expressions by collating similar terms. This is especially useful when dealing with expressions that include square roots.
Like terms have identical variable parts. For example, \( 3b\sqrt{2} \) and \( 20b\sqrt{2} \) are like terms, as they both contain \( b\sqrt{2} \).Combining these terms involves adding or subtracting the coefficients of these like terms. In the example of \( 3b\sqrt{2} + 20b\sqrt{2} \), you simply add the coefficients like so: 3 plus 20, which gives you 23.
Therefore, the expression simplifies to \( 23b\sqrt{2} \).Here's a concise process to follow:

• Identify the common parts in each term.
• Ensure the coefficients are added together.
• Rewrite the expression with the new coefficient.

This approach keeps your work organized and reduces the potential for mistakes, leading you to the simplest form effortlessly.

Factoring Under the Radical

Factoring is a crucial step when simplifying square roots, as it helps to efficiently reduce expressions. When factoring under the radical, the aim is to deconstruct the radicand into smaller, manageable parts - preferably containing one or more perfect squares.
This makes it easier to simplify the square root.Take the number 800, for instance.
To factor it under the square root, notice that we can express it as \(400 \times 2\), assuming we know 400 is a perfect square. Thus, under the radical, 800 decomposes: \(\sqrt{800} = \sqrt{400 \times 2} = \sqrt{400} \cdot \sqrt{2} = 20\sqrt{2}\).The steps for factoring under the radical include:

• Identify factors that are perfect squares.
• Decompose the radicand accordingly.
• Simplify the expression by extracting the square roots of those perfect squares.

This method reduces complexity and aids in easily combining like terms later on.