Simplify the Radical Expression 2 square root of 5+5 square root of 5-4 square root of 5

Solution:

Step 1:

Combine the like terms $2 \sqrt{5}$ and $5 \sqrt{5}$ to get $7 \sqrt{5}$. Now, the expression is $7 \sqrt{5} - 4 \sqrt{5}$.

Step 2:

Next, subtract $4 \sqrt{5}$ from $7 \sqrt{5}$ to simplify the expression further, which results in $3 \sqrt{5}$.

Step 3:

The simplified expression can be presented in various formats. In its exact form, it is $3 \sqrt{5}$. If converted to decimal, it is approximately $6.70820393 \ldots$.

Knowledge Notes:

To simplify a radical expression with like terms, you follow these steps:

1. Identify like terms: In radical expressions, terms are "like" if they have the same radical part. In this case, all terms have $\sqrt{5}$, so they are like terms.

2. Combine like terms: This is similar to combining like terms in algebraic expressions. You add or subtract the coefficients (the numbers in front of the radical) while keeping the radical part the same.

3. Simplify the expression: After combining like terms, you simplify the expression to its simplest form.

4. Convert to decimal if necessary: Sometimes, it's useful to see the approximate decimal value of a radical expression. This can be done using a calculator.

In this problem, the radical expression $2 \sqrt{5} + 5 \sqrt{5} - 4 \sqrt{5}$ is simplified by combining like terms and then subtracting as indicated. Since all terms have the same radical part ($\sqrt{5}$), we can add and subtract the coefficients as if they were regular numbers. The result is an exact form, which can also be expressed as a decimal.