Problem 151 - Elementary Algebra

Question

In the following exercises, simplify. $$ 7 \sqrt{y}+2 \sqrt{y} $$

Step-by-Step Solution

Answer

The simplified expression is \(9 \, \sqrt{y}\).

Step 1 - Identify Like Terms

Observe that both terms, \(7 \, \sqrt{y}\) and \(2 \, \sqrt{y}\), have the square root of \(y\) as a common factor.

Step 2 - Combine Like Terms

Add the coefficients of the like terms together. The coefficients are \(7\) and \(2\). Perform the addition: \(7 + 2 = 9\).

Step 3 - Write the Simplified Expression

After adding the coefficients, multiply the result by the common factor \(\sqrt{y}\): \(9 \, \sqrt{y}\).

Step 4 - Final Answer

The simplified expression is \(9 \, \sqrt{y}\).

Key Concepts

Like Terms

In this exercise, it's important to recognize like terms. Like terms are terms that have the same variable(s) raised to the same power. They can be combined because they represent the same quantity. For example, in the expression given, both terms, \(7 \, \sqrt{y}\) and \(2 \, \sqrt{y}\), have the square root of \(y\) as their common factor. That means they are like terms. To combine them, simply add their coefficients and keep the variable part the same.

Coefficients

Coefficients are the numerical factors in terms that multiply the variable. In the given problem, the coefficients are the numbers \(7\) and \(2\) in front of \(\sqrt{y}\). To simplify expressions with like terms, add the coefficients together. For this exercise, you add \(7\) and \(2\) to get \(9\). Then, attach the sum back to the variable part, which is \(\sqrt{y}\). This gives you the simplified expression \(9 \, \sqrt{y}\).

Square Root

The square root symbol \( \sqrt{} \) represents a number that, when multiplied by itself, gives the original number. For instance, \( \sqrt{y} \) indicates a number that, when squared, equals \(y\). In the provided exercise, you are dealing with terms that include the square root of \(y\). Even though square roots can seem complicated at first, combining them is straightforward. As long as the square roots are the same, you just combine the coefficients as shown.