Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(15 \sqrt{5}+7 \sqrt{45}\)

Step-by-Step Solution

Answer

The simplified expression is \(36 \sqrt{5}\).

Step 1: Identify Like Terms

In the expression \(15 \sqrt{5} + 7 \sqrt{45}\), notice that both terms have square roots. We need to check if these terms can be combined by simplifying the square roots so they have the same radicand.

Step 2: Simplify the Square Roots

Simplify \(\sqrt{45}\) as it can be broken down further. The number 45 can be factored into \(9 \times 5\) and since 9 is a perfect square, \(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3 \sqrt{5}\).

Step 3: Rewrite the Expression

Now rewrite the original expression using the simplified square root: \(15 \sqrt{5} + 7(3 \sqrt{5})\).

Step 4: Simplify the Expression

Multiply the terms: \(7 \times 3 \sqrt{5} = 21 \sqrt{5}\). Thus, the expression becomes \(15 \sqrt{5} + 21 \sqrt{5}\).

Step 5: Combine Like Terms

Now that both terms have the same square root \(\sqrt{5}\), combine them by adding the coefficients: \(15 + 21 = 36\). Thus, \(15 \sqrt{5} + 21 \sqrt{5} = 36 \sqrt{5}\).

Key Concepts

Combining Like Terms

Combining like terms is a key technique in simplifying expressions. It involves simplifying expressions by merging terms that share identical variables and exponents.

• For instance, in the expression, the terms \(15 \sqrt{5}\) and \(21 \sqrt{5}\) are like terms because they share a common variable - the square root of 5.
• When terms are like, you only need to add or subtract their coefficients. Here, the coefficients are 15 and 21.
• Thus, when you combine \(15 \sqrt{5}\) and \(21 \sqrt{5}\), you simply add 15 and 21 to obtain \(36 \sqrt{5}\).

By combining like terms, you streamline expressions, making them easier to understand and compute. It’s like merging similar items together for efficiency.

Square Roots

Understanding square roots is pivotal in simplifying expressions with radicals. A square root is a value that, when multiplied by itself, gives the original number.

• For example, the square root of 25 is 5 since \(5 \times 5 = 25\).
• Square roots are commonly represented using the radical symbol \(\sqrt{}\).
• In algebra, not all numbers have a whole number as their square root, making simplification a bit more challenging.

In the exercise, the square root of 45 needed further simplification because 45 can be broken down into factors of 9 and 5. Since 9 is a perfect square, you can simplify \(\sqrt{45}\) to \(3 \sqrt{5}\). By simplifying square roots, expressions become clearer and easier to manage.

Perfect Squares

Understanding perfect squares can greatly assist in simplifying square roots. A perfect square is an integer that is the square of another integer.

• Common perfect squares include numbers like 1, 4, 9, 16, 25, and so on.
• Recognizing perfect squares helps in breaking down square roots into simpler forms.
• For instance, in the expression \(\sqrt{45}\), recognizing that 9 is a perfect square allows this to be simplified to \(3 \sqrt{5}\).

This knowledge not only aids in simplifying square roots but also plays a role in various mathematical disciplines, like algebra, geometry, and calculus. Mastering perfect squares can make calculations faster and more accurate.