Problem 40 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \sqrt{7}(\sqrt{6}-\sqrt{3}) $$

Step-by-Step Solution

Answer

Question: Simplify the expression: \(\sqrt{7}(\sqrt{6} - \sqrt{3})\) Answer: \(\sqrt{42} - \sqrt{21}\)

Step 1 - Distribute the square root of 7

To distribute the square root of 7, multiply it with each term inside the parentheses: $$ \sqrt{7}(\sqrt{6}-\sqrt{3}) $$ After distribution, we get: $$ \sqrt{7}\cdot\sqrt{6} - \sqrt{7}\cdot\sqrt{3} $$

Step 2 - Simplify the expression

Now, we can use the property of square roots, that states \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\). So we have: $$ \sqrt{7\cdot6} - \sqrt{7\cdot3} $$ Now, we can simplify the expression further: $$ \sqrt{42} - \sqrt{21} $$ Since we cannot simplify any further, the final answer would be: $$ \boxed{\sqrt{42} - \sqrt{21}} $$

Key Concepts

Understanding the Distributive Property

Let's dive into the distributive property—a cornerstone concept in algebra. In the context of simplifying radicals, this property plays a crucial role. The distributive property allows us to multiply a single term with each term inside a parenthesis separately. In simpler terms, it is like spreading out multiplication over addition or subtraction inside parentheses. For example:

• Consider the expression: \( a(b+c) \).
• Using the distributive property, we rewrite it as: \( ab + ac \).

In our problem, \( \sqrt{7}(\sqrt{6} - \sqrt{3}) \), we distribute \( \sqrt{7} \) into the expression inside the parentheses. This means multiplying \( \sqrt{7} \) with both \( \sqrt{6} \) and \( \sqrt{3} \) separately. Recognizing when and how to use this property can make simplifying radical expressions much easier.

Exploring the Properties of Square Roots

Square roots have their own set of rules that help simplify calculations. One of the most important properties is that the product of two square roots is equal to the square root of the product of the numbers:

• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \)

This property essentially allows you to combine or separate square roots when multiplying numbers. In the original solution, it was used to transform \( \sqrt{7} \cdot \sqrt{6} \) into \( \sqrt{42} \).

Another useful property to remember is that the square root of a number squared gives the number itself: \( \sqrt{a^2} = a \). While not directly used in our specific solution, keeping these properties in mind is invaluable when simplifying more complex expressions. Mastering these properties allows for the easy transformation and simplification of radical expressions.

Effortless Multiplication of Radicals

Multiplying radicals can seem daunting, but with a few simple steps, it becomes manageable. The key is recognizing when to apply multiplication properties efficiently. When multiplying radicals, always look to apply the rule of multiplying them under one radical sign. This means combining the numbers first and then taking the square root of the result.

• For instance, \( \sqrt{7} \cdot \sqrt{6} \) becomes \( \sqrt{42} \), resulting from \( 7 \cdot 6 = 42 \).

Breaking it down:

• Multiply the numbers: \( a \times b \).
• Place the product under a single square root: \( \sqrt{ab} \).

This method simplifies the expression and minimizes errors, especially when dealing with more than one radical. Practice will make this approach second nature, allowing you to quickly deal with radicals in your algebra problems.