Problem 145 - Elementary Algebra

Question

In the following exercises, simplify. $$ 8 \sqrt{2}-5 \sqrt{2} $$

Step-by-Step Solution

Answer

3 \(\root2\)

Step 1: Identify Like Terms

In the expression, identify the like terms. Both terms contain the square root of 2, which means they can be combined.

Step 2: Combine the Coefficients

Add or subtract the coefficients of the like terms. Here, the coefficients are 8 and -5. Calculate 8 - 5.

Step 3: Multiply the Result by the Common Radical

Once the coefficients are combined, multiply the result by the common radical, which is \(\root2\). This gives you (8 - 5) \(\root2\).

Step 4: Final Simplified Expression

Calculate the final simplified expression. (8 - 5) \(\root2\) equals 3 \(\root2\).

Key Concepts

like terms

Like terms are terms in an algebraic expression that have the same variable raised to the same power. This means you can add or subtract them easily. For instance, in our exercise, both terms have the square root of 2, written as \(\root2\). They are considered like terms because the variable part, \(\root2\), is the same in each term. When you identify like terms, you can then combine them. This makes the expression simpler and easier to work with.

combining coefficients

Once you've identified like terms, the next step is to combine their coefficients. Coefficients are the numerical parts of the terms. In the example \(8 \root2 - 5 \root2\), the coefficients are 8 and -5. To simplify the expression, subtract the smaller coefficient from the larger one: \(8 - 5 = 3\). After combining the coefficients, you attach the common radical part back to the result. So, \(8 \root2 - 5 \root2\) simplifies to \(3 \root2\).

square roots

Understanding square roots is essential when simplifying radical expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because \(2 \times 2 = 4\). In our exercise, we're dealing with \(\root2\). No need to change the radical part when we combine terms, just focus on the coefficients. So, whether you have \(8 \root2\) or \(-5 \root2\), the \(\root2\) part remains constant throughout the simplification process.