Problem 75 - Algebra for College Students

Question

Is the expression \(3 \sqrt{2}+\sqrt{50}\) in simplest radical form? Defend your answer.

Step-by-Step Solution

Answer

Yes, \(3\sqrt{2} + \sqrt{50} = 8\sqrt{2}\) is in simplest radical form.

Step 1: Understanding Simplest Radical Form

An expression is said to be in its simplest radical form when all the radicals are simplified, and no radicals appear in the denominator. To simplify a radical, the number inside the square root should be factored such that any perfect squares can be removed from under the radical.

Step 2: Simplify \( \sqrt{50} \)

The number 50 can be factored into 25 and 2, where 25 is a perfect square. Thus, \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2} \).

Step 3: Substitute and Combine Terms

Substitute \( \sqrt{50} \) with its simplified form in the expression. The expression becomes \( 3\sqrt{2} + 5\sqrt{2} \). Since both terms are like terms (both have \( \sqrt{2} \)), you can combine them: \( (3 + 5)\sqrt{2} = 8\sqrt{2} \).

Step 4: Confirm the Simplest Radical Form

The expression \( 8\sqrt{2} \) is in its simplest radical form as 8 is an integer, and there are no further perfect square factors within the radical.

Key Concepts

Simplest Radical Form

Simplifying an expression into its simplest radical form means making sure the radicand—the number under the square root—doesn't contain any perfect square factors.

• A perfect square is a number that can be expressed as the product of an integer with itself, such as 4, 9, 16, etc.
• When an expression is in simplest radical form, all possible perfect squares are factored out, leaving only non-perfect square factors within the radical.
• Additionally, the expression should not have radicals in the denominator.

For example, consider the expression \( \sqrt{50} \). The number 50 can be broken down as \( 25 \times 2 \), where 25 is a perfect square. Therefore, \( \sqrt{50} \) simplifies to \( \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \). This step ensures there are no perfect squares inside the radical.

Factoring Radicals

Factoring radicals involves expressing the number inside the radical as a product of its prime factors, simplifying whenever possible.

• To simplify \( \sqrt{n} \), first find if \( n \) can be divided into factors, where one of them is a perfect square.
• Identify the largest perfect square factor to simplify efficiently.
• Extract the square root of the perfect square factor outside the radical. This reduces the complexity under the radical symbol.

In the exercise example, \( \sqrt{50} \) is decomposed into \( \sqrt{25 \times 2} \), then further simplified to \( 5 \sqrt{2} \). Understanding how to factor numbers into prime powers allows you to identify which radicals can be rewritten in simpler forms.

Combining Like Terms

Combining like terms is a crucial step in simplifying radical expressions. This process involves adding or subtracting terms that have the identical radicand part.

• Like terms contain the same variable factor or literal part, which in the case of radicals means the same square root part.
• Only coefficients are affected when combining like terms. The radical part remains unchanged.

In the expression \( 3\sqrt{2} + 5\sqrt{2} \), both terms have \( \sqrt{2} \) as their radicand, making them like terms. Thus, you combine them by adding their coefficients, resulting in \( 8\sqrt{2} \). This is how you effectively simplify the sum of radicals.