Problem 147 - Elementary Algebra

Question

In the following exercises, simplify. $$ 3 \sqrt{5}+6 \sqrt{5} $$

Step-by-Step Solution

Answer

9 \(\sqrt{5}\)

Step 1: Identify Like Terms

In the expression, notice that both terms contain a square root of 5: \(3 \sqrt{5}\) and \(6 \sqrt{5}\). These are like terms because they have the same radical part.

Step 2: Add the Coefficients

Since the terms are like terms, add their coefficients together. The coefficients are 3 and 6: \(3 + 6 = 9\).

Step 3: Recombine the Result

Combine the sum of the coefficients with the common square root of 5: \(9 \sqrt{5}\).

Key Concepts

Like Terms

In algebra, like terms are terms that contain the same variables raised to the same power. When working with expressions involving square roots, like terms are those that have the same radical part.
For example, in the expression you provided, both terms have the square root of 5: \(3 \sqrt{5}\) and \(6 \sqrt{5}\). Because they share the same square root, they are considered like terms.
This makes it much easier to simplify the expression because you can combine these terms just like you would with regular numbers or variables.

Coefficients

Coefficients are the numerical part of terms that include variables or radicals. They tell you how many of each term you have.
In the expression \(3 \sqrt{5} + 6 \sqrt{5}\), the coefficients are 3 and 6. When you have like terms, you add or subtract their coefficients while keeping the variable or radical part the same.
So, in our exercise, you add the coefficients 3 and 6:
\[3 + 6 = 9\]
This sums up to give us a new coefficient for the term involving \(\sqrt{5}\).

Square Roots

A square root is a special type of radical expression. It represents a value that, when multiplied by itself, gives the original number.
For example, \(\sqrt{5}\) means a number that, when squared, equals 5. In simplified radical expressions, you often encounter like terms that have the same square root.
In your exercise, combining \(3 \sqrt{5}\) and \(6 \sqrt{5}\) results in a single term with the same square root but a combined coefficient:
\[3 \sqrt{5} + 6 \sqrt{5} = 9 \sqrt{5}\]
This keeps the expression simple and easy to understand.