Question

Simplify each expression. Assume that all variables are positive when they appear. $$6 \sqrt{5}-4 \sqrt{5}$$

Short answer

2 \sqrt{5}\

Step-by-step solution

Identify Like Terms

Recognize that both terms have \( \sqrt{5} \) which means they are like terms.

Combine Coefficients

Add or subtract the coefficients of the like terms. For \(6 \sqrt{5} - 4 \sqrt{5} \), subtract 4 from 6.

Simplify the Expression

After combining the coefficients, \(6 - 4 = 2\), so the simplified expression is \(2 \sqrt{5}\).

Key concepts

Like Terms

In algebra, like terms are terms that have the same variables and powers. For instance, in the expression 6\( \sqrt{5} \) and -4\( \sqrt{5} \), both terms include the square root of 5. This makes them like terms.
Recognizing like terms is crucial because it allows you to simplify expressions by combining them.
When terms share the same variables and exponents, we can add or subtract them directly.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 * 5 = 25.
In the given problem, we are dealing with the square root of 5, represented as \( \sqrt{5} \).
Square roots can sometimes make expressions look complex, but they follow the same rules as other terms when identifying like terms and combining coefficients.

Combining Coefficients

Coefficients are the numerical parts of terms that include variables or roots. In the expression 6\( \sqrt{5} - 4\( \sqrt{5} \)\), the coefficients are 6 and -4, respectively.
To combine the coefficients, simply add or subtract them as indicated in the expression. This does not change the square root, it only affects the numerical part.

• Step 1: Identify the coefficients: 6 and -4.
• Step 2: Subtract -4 from 6: 6 - 4 = 2.
• Step 3: Attach the square root back with the new coefficient: 2\( \sqrt{5} \).

Combining coefficients is a crucial step in simplifying expressions involving like terms, as it directly affects the final simplified form.