Problem 153 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{a}-4 \sqrt{a} $$

Step-by-Step Solution

Answer

-3 \sqrt{a}

Step 1: Identify like terms

Look at the expression \( \sqrt{a} - 4\sqrt{a} \). Notice that both terms contain the radical \( \sqrt{a} \), making them like terms.

Step 2: Combine like terms

Since both terms are like terms, you can combine them by subtracting the coefficients. The expression turns into \( (1-4)\sqrt{a} \).

Step 3: Simplify the expression

Simplify \( (1-4)\sqrt{a} \) to get \ -3\sqrt{a} \.

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 \( \sqrt{a} - 4\sqrt{a} \), both terms contain the variable \( \sqrt{a} \). This means they are like terms.

Like terms can be easily combined because they share the same variable part. This is important because it allows us to simplify expressions by adding or subtracting their coefficients. Identifying like terms is the first step in simplifying any algebraic expression. It tells us which parts of the expression can be combined.

combining coefficients

Combining coefficients is the process of adding or subtracting the numerical parts of like terms. In the expression \( \sqrt{a} - 4\sqrt{a} \), the coefficients are 1 and -4. These numbers are the coefficients associated with the radical term \( \sqrt{a} \).

To combine them, we perform the operation given in the expression:

- Since we have 1\( \sqrt{a} \) and we're subtracting 4\( \sqrt{a} \), we calculate \( 1 - 4 \) to get \( -3 \).

Thus, \( \sqrt{a} - 4\sqrt{a} \) simplifies to \( -3\sqrt{a} \).

radical expressions

Radical expressions are mathematical expressions that include a square root, cube root, or any higher root. In our example, \( \sqrt{a} \) is a square root and is considered a radical expression.

Simplifying radical expressions usually involves combining like terms (if possible) and simplifying the coefficients. In the expression \( \sqrt{a} - 4\sqrt{a} \), we dealt with like terms and combined their coefficients to finally get \( -3\sqrt{a} \. \).

Understanding how to work with radicals is key for algebra and higher levels of math. Always remember to look for terms with the same radical part and then operate on their coefficients. This makes simplifying such expressions straightforward.