Problem 60 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ \sqrt{3}(\sqrt{3}-2 \sqrt{5 x}) $$

Step-by-Step Solution

Answer

The simplified expression is \(3 - 2\sqrt{15x}\).

Step 1: Distribute the Square Root

We need to distribute \( \sqrt{3} \) to both terms inside the parentheses. This means we'll multiply \( \sqrt{3} \) by each term one at a time: \( \sqrt{3} \times \sqrt{3} \) and \( \sqrt{3} \times (-2\sqrt{5x}) \).

Step 2: Simplify Each Term

First, simplify \( \sqrt{3} \times \sqrt{3} \). Since \( \sqrt{3} \times \sqrt{3} = 3 \), this expression simplifies to 3.Next, simplify \( \sqrt{3} \times (-2\sqrt{5x}) \). This is equivalent to \(-2\times \sqrt{15x}\), as multiplying square roots gives us \( \sqrt{3} \times \sqrt{5x} = \sqrt{15x} \).

Step 3: Write the Simplified Expression

Combine the simplified terms from Step 2. The expression becomes:\[ 3 - 2\sqrt{15x} \].

Key Concepts

Simplifying Expressions

When you simplify an expression, you are making it easier to understand and work with. Simplification involves manipulating the expression to its simplest form without changing its value.
One important thing to remember is to combine like terms. For example, if you have similar root expressions or constants, you can try to add or subtract them.
In algebra, simplifying means reducing expressions by canceling out terms or factoring them. Make sure to perform operations in a way that gets rid of any complexity while maintaining accuracy.

Square Root

A square root is a number that produces a specified quantity when multiplied by itself. Think of it as the opposite of squaring a number. For instance, the square root of 9 is 3 because 3 multiplied by itself equals 9.
When dealing with square roots, you may encounter expressions involving \(\sqrt{a}\sqrt{b}\), where a and b are non-negative. If possible, try to simplify this as \(\sqrt{ab}\), which means multiplying the numbers under the radical sign. Understanding how a square root functions is pivotal for simplifying complex algebraic expressions that involve radicals.

Distributive Property

The distributive property helps us break down expressions into simpler parts. It's particularly helpful for expressions inside parentheses. The formula is: \(a(b + c) = ab + ac\).
In our example, we distributed \(\sqrt{3}\) to every term inside the parentheses: \(\sqrt{3}\sqrt{3}\) and \(\sqrt{3}(-2\sqrt{5x})\). After distributing, you simplify each term individually.
Understanding this property ensures that you don't overlook parts of the expression that need multiplying. It's a fundamental concept for working through many algebra problems.

Multiplication of Radicals

Multiplying radicals involves multiplying the numbers under the square root sign first. In our example, we multiplied \(\sqrt{3}\sqrt{5x}\) to get \(\sqrt{15x}\). Remember these key points:

• Multiply the coefficients outside the radicals.
• Multiply the numbers inside the radicals.

The results form a new chain under a single square root whenever possible. Simplification is easier if you understand that \(\sqrt{a} * \sqrt{b} = \sqrt{ab}\) as it cuts down complexity and helps in clarifying results.