Problem 56 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. $$ 4+\sqrt{8}+\sqrt{2}+8 $$

Step-by-Step Solution

Answer

The simplified expression is \( 12 + 3\sqrt{2} \).

Step 1: Identify and Simplify Radicals

First, look for radicals that can be simplified. Notice that \( \sqrt{8} \) can be simplified. Since \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \), we replace \( \sqrt{8} \) with \( 2\sqrt{2} \).

Step 2: Rewrite the Expression

Substitute the simplified radical back into the expression. The expression becomes \( 4 + 2\sqrt{2} + \sqrt{2} + 8 \).

Step 3: Combine Like Terms

Now, combine the constants and like radicals. First, add the constant terms: \( 4 + 8 = 12 \). Next, combine the like radicals \( 2\sqrt{2} + \sqrt{2} = 3\sqrt{2} \).

Step 4: Form the Final Expression

Now combine the results of both the constants and the radicals. The expression simplifies to \( 12 + 3\sqrt{2} \).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves merging terms that have the same variable and exponent into a single term. In the context of simplifying radicals, we often encounter like terms that appear as similar radicals, meaning they share the same radicand.

• To combine, check if the radicals have the same number under the square root sign (the radicand).
• For example, with terms like \( \sqrt{2} \) and \( 2\sqrt{2} \), you can add them together since they share the same radicand: \( \sqrt{2} + 2\sqrt{2} = 3\sqrt{2} \).

This mirrors how you would handle combining terms like \( x \) and \( 2x \) in basic algebra, where they sum to \( 3x \). Recognizing like radicals allows you to simplify expressions efficiently and avoid unnecessary complexity.

Simplifying Expressions

Simplifying expressions refers to the process of rewriting expressions in their simplest form. Simplification often involves reducing the number of terms and operations required to evaluate an expression.

• This can include using operations to combine like terms, simplifying fractions, or breaking down more complex mathematical forms into simpler ones.
• In the context of radicals, simplifying means expressing each radical in a form that cannot be simplified further.

For example, the expression \( \sqrt{8} \) can be simplified because 8 can be factored into 4 and 2, and 4 is a perfect square. Thus, \( \sqrt{8} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \). Simplification not only makes expressions easier to manage but also helps in further operations like addition or multiplication.

Intermediate Algebra Concepts

Intermediate algebra concepts build upon basic algebra and include a deeper understanding of expressions, equations, and functions. Mastery in simplifying expressions and combining like terms is crucial at this level.

• This involves dealing with more complex terms and expressions, often involving variables, exponents, and radicals.
• An important skill is radical manipulation, which includes simplifying, adding, subtracting, multiplying, and sometimes even rationalizing denominators.

Consider an expression like \( 4 + \sqrt{8} + \sqrt{2} + 8 \). First, simplify any radicals as learned: \( \sqrt{8} = 2\sqrt{2} \). Substitute this back, resulting in \( 4 + 2\sqrt{2} + \sqrt{2} + 8 \). Then, you apply the skill of combining like terms by adding constants and like radicals: \( 4 + 8 = 12 \) and \( 2\sqrt{2} + \sqrt{2} = 3\sqrt{2} \), which simplifies to \( 12 + 3\sqrt{2} \).
This showcases not just complex arithmetic but bolsters your foundational understanding critical for higher algebra.