Problem 149 - Elementary Algebra

Question

In the following exercises, simplify. $$ 9 \sqrt{7}-10 \sqrt{7} $$

Step-by-Step Solution

Answer

-\sqrt{7}

Step 1: Identify like terms

Both terms in the expression involve the square root of 7. This means they are like terms and can be combined.

Step 2: Combine like terms

Combine the coefficients of the square root of 7. Subtract 10 from 9, keeping the square root of 7 common: \[ 9\sqrt{7} - 10\sqrt{7} \]

Step 3: Simplify the expression

Perform the subtraction by combining the coefficients: \[ 9 - 10 \] This results in: \[ -1\]. Thus the expression simplifies to: \[ -\sqrt{7} \]

Key Concepts

like terms

In algebra, like terms are terms that contain the same variable raised to the same power. For example, in the expression \(9\sqrt{7} - 10\sqrt{7}\), both terms include the square root of 7. This makes them like terms because the variable part of both terms (\(\sqrt{7}\)) is identical. Like terms allow us to combine them through addition or subtraction by simply focusing on their numerical coefficients.
When simplifying expressions, identifying like terms is crucial. It allows us to streamline complex expressions, making them easier to work with.

coefficients

A coefficient is the numerical part of a term that contains a variable. Using our example \(9\sqrt{7} - 10\sqrt{7}\), the coefficients are 9 and -10. The square root of 7 remains consistent across both terms, and our task is to combine the coefficients.
To do this, simply subtract the second coefficient from the first: \(9 - 10 = -1\). So, the original expression collapses to \(-\sqrt{7}\). By handling the coefficients correctly, we simplify the term without affecting the variable part.

subtraction with radicals

When subtracting radical expressions, the key is to ensure the radicals are the same, meaning they are like terms. For example, in \(9\sqrt{7} - 10\sqrt{7}\), both terms share the \(\sqrt{7}\) component.
By subtracting the coefficients, we simplify the expression: \(9 - 10 = -1\). Therefore, \(9\sqrt{7} - 10\sqrt{7} = -\sqrt{7}\).
The process mirrors simple arithmetic but requires constant attention to the radicals involved. Simplifying correctly ensures accurate results in expressions with radicals.