Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{2}+5 \sqrt{2} $$

Step-by-Step Solution

Answer

The simplest form is \( 6\sqrt{2} \).

Step 1: Identify Like Terms

In the expression \( \sqrt{2}+5\sqrt{2} \), both terms are like terms because they have the same radical part \( \sqrt{2} \). Like terms can be combined by adding their coefficients.

Step 2: Combine Like Terms

To combine the like terms, add their coefficients. The coefficients are 1 (implied in \( \sqrt{2} \)) and 5 from the expression given. Therefore, \( 1\sqrt{2} + 5\sqrt{2} = (1 + 5)\sqrt{2} = 6\sqrt{2} \).

Step 3: Write the Simplified Expression

The simplest form of the expression \( \sqrt{2}+5\sqrt{2} \) is \( 6\sqrt{2} \).

Key Concepts

Like Terms

When working with mathematical expressions, the concept of 'like terms' is essential. Like terms are terms in an expression that have the same variable parts raised to the same power. For example, in an algebraic expression, terms like \( 3x \) and \( 5x \) are like terms because they both contain the same variable \( x \).In the context of radicals, two radical expressions are considered like terms if they have the same radicand (the part beneath the radical sign) and the same index (the type of root). For example, \( \sqrt{2} \) and \( 5\sqrt{2} \) are like terms because both terms have the radicand \( 2 \). This means that they can be combined by adding or subtracting their coefficients, just like regular algebra terms.For students, understanding like terms is a critical step in simplifying complex expressions. It allows you to correctly group terms and make calculations easier. Always look for common radicals or variable parts when trying to identify like terms.

Radicals

Radicals are mathematical expressions that involve roots, such as square roots, cube roots, etc. The term 'radical' refers to the radical symbol (\( \sqrt{\ } \)), which indicates the root of a number or expression. The number under the radical sign is called the radicand, and this tells you which number you're finding the root of.For example, consider the expression \( \sqrt{2} \). Here, the radical sign denotes that we're looking for the square root of \( 2 \). Radical expressions can often be simplified or combined with like terms, as we see in exercises involving radicals.Simplifying radicals involves reducing the radicand by taking out any perfect squares (in the case of square roots) or other perfect powers. However, in addition to simplifying, we can also combine radical expressions when they are like terms. The key takeaway is to always try to simplify the radicand first, and then look for opportunities to combine like terms where possible.

Coefficients

Coefficients are the numbers or constants that are multiplied by the variable parts of terms in expressions. In an algebraic term like \( 5x \), the number \( 5 \) is the coefficient, indicating how many times the variable \( x \) is being considered.When dealing with radicals, the coefficient is the number that multiplies the radical part. For example, in the term \( 5\sqrt{2} \), the number \( 5 \) is the coefficient of the \( \sqrt{2} \) term.Understanding coefficients is crucial for combining like terms because the coefficients of like terms are the parts that you directly add or subtract to simplify expressions. Taking our radical expression \( \sqrt{2}+5\sqrt{2} \), it is the coefficients 1 (for \( \sqrt{2} \)) and 5 (for \( 5\sqrt{2} \)) that we sum up. This leads us to the simplified form \( 6\sqrt{2} \). Always remember to keep track of coefficients when attempting to simplify or combine terms in any expression.