Problem 12 - Elementary Algebra

Question

For the following problems, simplify each of the square root expressions. $$ \sqrt{10}(\sqrt{8}-\sqrt{2}) $$

Step-by-Step Solution

Answer

Answer: \(2\sqrt{5}\)

Step 1: Simplify the square roots inside the parentheses

First, let's simplify the square roots inside the parentheses. Recall that \(\sqrt{ab} = \sqrt{a}\sqrt{b}\). In this case, it means that we need to find the prime factorization of the numbers inside the square roots and then pair the common factors for simplification. For \(\sqrt{8}\): $$ \sqrt{8} = \sqrt{2^3} = 2\sqrt{2} $$ For \(\sqrt{2}\): $$ \sqrt{2} = \sqrt{2^1} = 1\sqrt{2} $$ Now, let's substitute these simplified square roots into the original expression: $$ \sqrt{10}(2\sqrt{2} - 1\sqrt{2}) $$

Step 2: Distribute the square root outside the parentheses

Next, we will distribute \(\sqrt{10}\) to both simplified terms inside the parentheses: $$ \sqrt{10}(2\sqrt{2}) - \sqrt{10}(1\sqrt{2}) = 2\sqrt{20} - \sqrt{20} $$

Step 3: Simplify the resulting square roots

Now, let's simplify the new square roots: For \(\sqrt{20}\): $$ \sqrt{20} = \sqrt{2^2 \cdot 5} = 2\sqrt{5} $$ Substitute this simplified square root back into the expression: $$ 2\sqrt{20} - \sqrt{20} = 2(2\sqrt{5}) - 2\sqrt{5} $$

Step 4: Combine like terms

Finally, we can combine the like terms to simplify the expression: $$ 2(2\sqrt{5}) - 2\sqrt{5} = 4\sqrt{5} - 2\sqrt{5} = 2\sqrt{5} $$ So the simplified expression is: $$ \sqrt{10}(\sqrt{8}-\sqrt{2}) = 2\sqrt{5} $$

Key Concepts

Prime Factorization

Prime factorization involves breaking down a number into its smallest prime factors. This technique is essential when simplifying square roots, especially because it helps identify which factors can be taken out of the square root.

To find the prime factorization, start by dividing the number by the smallest prime (usually 2) and continue the process until you cannot divide further:

• For 8: The prime factors are 2, 2, and 2 because 8 can be written as \(2^3\). This means, \(\sqrt{8} = \sqrt{2^3} = 2\sqrt{2}\).
• For 20: The prime factors are 2, 2, and 5, thus \(20 = 2^2 \times 5\). Therefore, \(\sqrt{20} = \sqrt{2^2 \times 5} = 2\sqrt{5}\).

By identifying the prime factorization, we can easily determine which factors will be left inside the square root and which will come outside.

Distributive Property

The distributive property is a fundamental algebraic property used to simplify expressions and equations. It allows us to multiply a number by a group of terms inside parentheses. In simple terms, you "distribute" the multiplication across the terms inside:

In our example, we use the distributive property to simplify:

• \( \sqrt{10} (2\sqrt{2} - 1\sqrt{2})\)

Here, \(\sqrt{10}\) is multiplied by each term inside the parentheses:

• \(\sqrt{10} \times 2\sqrt{2} = 2\sqrt{20}\)
• \(\sqrt{10} \times 1\sqrt{2} = \sqrt{20}\)

After applying the distributive property, we can further simplify our result, making the entire expression easier to handle.

This property is not only limited to numbers but is also essential when dealing with algebraic equations, enabling us to manipulate and solve them efficiently.

Combining Like Terms

Combining like terms is a technique used in algebra to simplify expressions by merging terms that have the same variable parts. Like terms are terms whose variables (and their exponents) are exactly alike. This makes it easier to perform simple mathematical operations on them.

In the expression \(2(2\sqrt{5}) - 2\sqrt{5}\), we can see that both terms include \(\sqrt{5}\):

• First term: \(4\sqrt{5}\)
• Second term: \(2\sqrt{5}\)

By subtracting the second term from the first, we efficiently combine them:

• \(4\sqrt{5} - 2\sqrt{5} = 2\sqrt{5}\)

Combining like terms not only streamlines the expression but also often reveals the simplest form of a solution. This technique is crucial when solving equations or simplifying expressions, as it reduces complexity and provides clarity.