Problem 60 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ \sqrt{3}(\sqrt{3}-2 \sqrt{5 x}) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Term

Distribute the term \( \sqrt{3} \) to each term inside the parentheses: \[ \sqrt{3} \times (\sqrt{3} - 2 \sqrt{5x}) = \sqrt{3} \times \sqrt{3} - \sqrt{3} \times 2 \sqrt{5x} \]

Step 2: Simplify Each Product

Simplify each of the distributed products. - For \( \sqrt{3} \times \sqrt{3} \), use the property \( \sqrt{a} \times \sqrt{a} = a \): \[ \sqrt{3} \times \sqrt{3} = 3 \]- For \( \sqrt{3} \times 2 \sqrt{5x} \), multiply the coefficients first and keep \( \sqrt{3} \) and \( \sqrt{5x} \) under the same square root: \[ 2 \times \sqrt{3} \times \sqrt{5x} = 2 \sqrt{15x} \]

Step 3: Combine Simplified Terms

Combine the simplified terms into a single expression:\[ 3 - 2\sqrt{15x} \]

Step 4: Verify Simplification

Check if further simplification is possible. In this case, no further simplification is possible, as there are no like terms or simple factors that can be reduced. Thus, the expression is simplified as:\[ 3 - 2\sqrt{15x} \]

Key Concepts

The Distributive Property

The distributive property is a fundamental concept in mathematics. It allows us to simplify expressions by distributing one term across another group of terms inside parentheses. Imagine you have a basket of apples and you want to give different numbers of apples to two friends. Distributing would mean giving each friend some apples based on what's available in the basket.In mathematical terms, the distributive property works like this:

• When you have an expression of the form \(a(b + c)\), you can distribute \(a\) to both \(b\) and \(c\).
• This means it becomes \(ab + ac\).

Applying this to radical expressions, such as \(\sqrt{3}(\sqrt{3} - 2 \sqrt{5x})\), involves distributing \(\sqrt{3}\) over the terms inside the parentheses. First multiply \(\sqrt{3}\) by \(\sqrt{3}\), and then \(\sqrt{3}\) by \(-2 \sqrt{5x}\). This step sets the stage for further simplifications and is essential for handling expressions that include radicals.

Multiplication of Radicals

Multiplying radicals might feel tricky at first, but it follows a straightforward process. Radicals are symbols that represent roots, like square roots, and when you multiply them, there are some useful rules that help simplify the process.Here are some key points to remember:

• If you have \(\sqrt{a} \times \sqrt{b}\), you can combine them under a single radical: \(\sqrt{ab}\).
• Make sure to multiply the numbers (coefficients) outside the radicals first.

In our example, we first multiply \(\sqrt{3}\) by itself, using the rule \(\sqrt{a} \times \sqrt{a} = a\). This gives us \(3\). Then, for \(\sqrt{3} \times 2 \sqrt{5x}\), multiply the coefficients (numbers outside the square root), resulting in \(2\), and keep the multiplication of radicals under a single square root, giving \(2 \sqrt{15x}\). Understanding these basics boosts your confidence when dealing with more complex radical expressions.

Simplifying Expressions

Simplifying expressions is all about making them as neat and concise as possible. It involves combining like terms and making sure that every part of the expression is reduced to its simplest form.Here's what you need to know:

• Check if terms can be combined. Like terms often share a common base or exponent.
• If you're working with radicals, see if you can simplify the expression inside the radical.

In this exercise, after distributing and multiplying, we ended up with two terms: \(3 - 2\sqrt{15x}\). These terms are already in their simplest form since:

• There are no like terms to combine.
• Neither radical can be simplified further.

Verify your simplification to ensure nothing more can be done. Here, the expression \(3 - 2\sqrt{15x}\) is our final simplified result. Practicing these steps makes handling expressions much more straightforward and reduces errors in solving problems.