Problem 8 - Elementary Algebra

Question

Simplify each by performing the indicated operation. $$ (\sqrt{5}+\sqrt{7})(\sqrt{2}+\sqrt{8}) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((\sqrt{5}+\sqrt{7})(\sqrt{2}+\sqrt{8})\). Answer: The simplified expression is \(3\sqrt{10} + 3\sqrt{14}\).

Step 1: Apply distributive property

Multiply each term of the first expression (\(( \sqrt{5} + \sqrt{7})\)) by each term of the second expression (\(( \sqrt{2} + \sqrt{8})\)). This will give us: $$ (\sqrt{5} + \sqrt{7})(\sqrt{2} + \sqrt{8}) = \sqrt{5}(\sqrt{2}) + \sqrt{5}(\sqrt{8}) + \sqrt{7}(\sqrt{2}) + \sqrt{7}(\sqrt{8}) $$

Step 2: Perform the multiplication

Multiply the terms inside the square roots: $$ \sqrt{5}(\sqrt{2}) + \sqrt{5}(\sqrt{8}) + \sqrt{7}(\sqrt{2}) + \sqrt{7}(\sqrt{8}) = \sqrt{10} + \sqrt{40} + \sqrt{14} + \sqrt{56} $$

Step 3: Simplify the square roots

Simplify the square roots with perfect square factors: $$ \sqrt{10} + \sqrt{40} + \sqrt{14} + \sqrt{56} = \sqrt{10} + 2\sqrt{10} + \sqrt{14} + 2\sqrt{14} $$

Step 4: Combine like terms

Combine like terms and write the final simplified expression: $$ \sqrt{10} + 2\sqrt{10} + \sqrt{14} + 2\sqrt{14} = 3\sqrt{10} + 3\sqrt{14} $$ So the simplified form of the given expression is: $$ (\sqrt{5}+\sqrt{7})(\sqrt{2}+\sqrt{8}) = 3\sqrt{10} + 3\sqrt{14} $$

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical concept that states you can multiply a single term by each term in a parenthesis. It helps simplify expressions when you're multiplying two or more terms together.

In this exercise, we applied the distributive property to the expression \((\sqrt{5} + \sqrt{7})(\sqrt{2} + \sqrt{8})\). Here's how it works:

• Multiply each term in the first group by each term in the second group.
• We perform: \(\sqrt{5}(\sqrt{2}) + \sqrt{5}(\sqrt{8}) + \sqrt{7}(\sqrt{2}) + \sqrt{7}(\sqrt{8})\).

This allows us to reorganize and simplify terms further, leading to a much cleaner and more manageable expression.

Understanding and utilizing the distributive property is crucial as it forms the foundation for solving a variety of algebraic problems. It streamlines calculations by breaking down complex expressions into more straightforward components.

Multiplying Radicals

When multiplying radicals, it's important to remember that you can combine them under a single radical sign, as long as the indices are the same. This means if you multiply two square roots together, you can combine them, like so: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).

As part of our exercise, we multiplied the radicals to obtain:

• \(\sqrt{5}(\sqrt{2}) = \sqrt{10}\)
• \(\sqrt{5}(\sqrt{8}) = \sqrt{40}\)
• \(\sqrt{7}(\sqrt{2}) = \sqrt{14}\)
• \(\sqrt{7}(\sqrt{8}) = \sqrt{56}\)

After performing these multiplications, we move on to look for perfect square factors within these numbers to simplify further.

Multiplying radicals is a valuable skill, particularly when dealing with equations involving square roots or simplifying expressions to their most basic form. This step ensures each part of the expression is dealt with individually before combining similar terms later.

Combining Like Terms

Combining like terms is the final piece of the puzzle when simplifying expressions. In the context of radicals, like terms refer to expressions that have the same radical part. By recognizing and merging these similar parts, we simplify the expression.

Consider the following radicals from our exercise:

• \(\sqrt{10} + 2\sqrt{10}\) are like terms because both have \(\sqrt{10}\) as the radical part.
• \(\sqrt{14} + 2\sqrt{14}\) are also like terms with \(\sqrt{14}\) as the radical component.

To combine these, add the coefficients (numbers in front of the radicals), resulting in:

• \(\sqrt{10} + 2\sqrt{10} = 3\sqrt{10}\)
• \(\sqrt{14} + 2\sqrt{14} = 3\sqrt{14}\)

We conclude with the expression: \(3\sqrt{10} + 3\sqrt{14}\).

Learning to combine like terms helps tidy up expressions, making them easier to read and solve.