Problem 35 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ 4 a x \sqrt{3 x}+2 \sqrt{3 a^{2} x^{3}}+7 \sqrt{3 a^{2} x^{3}} $$

Step-by-Step Solution

Answer

The like terms in the expression are \(2 \sqrt{3 a^{2} x^{3}}\) and \(7 \sqrt{3 a^{2} x^{3}}\). The simplified form of the expression is \(4 a x \sqrt{3 x} + 9 \sqrt{3 a^{2} x^{3}}\).

Step 1: Identify Like Terms

The given expression is: $$ 4 a x \sqrt{3 x} + 2 \sqrt{3 a^{2} x^{3}} + 7 \sqrt{3 a^{2} x^{3}} $$ We can see that the terms \( 2 \sqrt{3 a^{2} x^{3}}\) and \(7 \sqrt{3 a^{2} x^{3}} \) are like terms.

Step 2: Group Like Terms

Group like terms in the given expression: $$ 4 a x \sqrt{3 x} + \left(2 \sqrt{3 a^{2} x^{3}} + 7 \sqrt{3 a^{2} x^{3}}\right) $$

Step 3: Combine Like Terms

Combine the like terms by adding their coefficients: $$ 4 a x \sqrt{3 x} + (2 + 7) \sqrt{3 a^{2} x^{3}} $$

Step 4: Simplify

Now, we just need to simplify the coefficients and write the final expression: $$ 4 a x \sqrt{3 x} + 9 \sqrt{3 a^{2} x^{3}} $$ So the simplified expression is: $$ 4 a x \sqrt{3 x} + 9 \sqrt{3 a^{2} x^{3}} $$

Key Concepts

Understanding Radicals

Radicals are at the heart of many algebraic expressions, and they often show up as a square root or another type of root, symbolized by the radical sign (√). In this exercise, you can spot radicals quite easily by looking for the √ symbol. For instance, in the expression \(4 a x \sqrt{3 x}\), \(\sqrt{3 x}\) is the radical part. A radical expression may sometimes seem complex, but it merely represents the root of a number or an expression.

Radicals have particular rules: they can only be combined or simplified if they have the same radicand (the number inside the radical). This is key when simplifying algebraic expressions with radicals, as it helps you identify like terms when they're present. Always keep an eye on both the coefficient (the number in front of the radical) and the radicand to determine how you can simplify terms with radicals.

Identifying and Grouping Like Terms

Like terms in algebra are terms that have the same variable raised to the same power. They allow you to simplify expressions because only like terms can be combined through addition or subtraction. In the given expression, you see terms like \(2 \sqrt{3 a^{2} x^{3}}\) and \(7 \sqrt{3 a^{2} x^{3}}\). These are considered like terms because they share the same radical expression \(\sqrt{3 a^{2} x^{3}}\).

When you group like terms, you can transparently see which terms can be combined. This is crucial because it simplifies the work drastically. By writing them next to each other, as in \(2 \sqrt{3 a^{2} x^{3}} + 7 \sqrt{3 a^{2} x^{3}}\), you prepare them for the next step: adding the coefficients.

The Addition of Coefficients

The process of coefficient addition is used to consolidate like terms into a simpler expression. In algebra, coefficients are the numbers that multiply the variables or radicals. In this context, when working with the terms \(2 \sqrt{3 a^{2} x^{3}} + 7 \sqrt{3 a^{2} x^{3}}\), you only need to add the numerical coefficients, which are 2 and 7, because they have like radical parts.

By focusing on this, you find that you can easily simplify the expression to \(9 \sqrt{3 a^{2} x^{3}}\). As seen, the radicands are identical, making it possible to sum up the coefficients directly. It's a straightforward numerical addition once the terms are correctly grouped, leading you to the final, simplified expression.