Problem 57 - Algebra 1: Concepts and Skills

Question

Simplify \(\sqrt{5}(6+\sqrt{5})\) $$ a.\quad \sqrt{30}+5 $$ $$ b.\quad 5 \sqrt{6}+5 $$ $$ c.\quad 6 \sqrt{5}+5 $$ $$ d.\quad 11 \sqrt{5} $$

Step-by-Step Solution

Answer

The simplified expression for \(\sqrt{5}(6+\sqrt{5})\) is option c. \(6\sqrt{5}+5\)

Step 1: Distribute the square root

Distribute the \(\sqrt{5}\) through the parentheses. This gives \(6\sqrt{5} + 5\). This is done by multiplying \(\sqrt{5}\) by 6 and then by \(\sqrt{5}\) itself.

Step 2: Simplify the square root of 5 squared

Simplify \(\sqrt{5} * \sqrt{5}\) to 5. This is based on the rule that when the same square root is multiplied by itself, it equals the square root number.

Step 3: Combine the numbers

Add together \(6\sqrt{5}\) and 5 to get the final simplified terms.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term within a parenthesis. Especially when simplifying radical expressions, it's important to distribute the radical correctly.

For instance, let's consider the expression \(\sqrt{5}(6+\sqrt{5})\). Here, you'll want to apply the distributive property by multiplying \(\sqrt{5}\) with 6 and then with \(\sqrt{5}\), separately. It's like sharing a piece of cake equally among friends. Each term inside the parenthesis gets an equal 'share' of \(\sqrt{5}\).

Being cautious while distributing ensures you won't miss any terms and will set the stage for further simplification using other algebraic rules.

Square Root Simplification

When dealing with square roots, simplification involves reducing expressions to their simplest form. It's like taking a complex puzzle and fitting the pieces together so that you can see the clear picture.

Square root simplification relies on the fact that the square root of a number squared is the number itself, as in our example where \(\sqrt{5} * \sqrt{5} = 5\). This reflects the basic rule of roots and powers: \(\sqrt{x^2} = x\), with the presumption that x is a non-negative number.

This concept helps to dramatically streamline expressions, making them more manageable and easier to combine with other like terms. In the exercise provided, understanding this crucial rule directly helps in step 2 of the solution.

Combining Like Terms

After distributing and simplifying square roots, the next step is often to combine like terms. This means to merge terms that have the same variable or the same radical component into a single term. Think of it as organizing a fruit salad; you group the same types of fruit together.

In our example, after simplification, you end up with \(6\sqrt{5}\) and 5. Since there are no like terms to combine further (as one term contains a radical and the other does not), this is typically the final step, giving you the sum of a radical and a whole number.

Mastering this process will help you to tidy up an algebraic expression, ensuring that it's presented in its simplest form. It's always worth double-checking after this step to confirm that no further simplification is possible.