Problem 32 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 3 \sqrt[3]{5}+4 \sqrt{5}-2 \sqrt{5} $$

Step-by-Step Solution

Answer

The simplified expression is \(3 \sqrt[3]{5} + 2 \sqrt{5}\).

Step 1: Identify Like Terms

Examine the expression to identify like terms. In the given expression, **\(3 \sqrt[3]{5}\)** stands alone as a cubic root, while **\(4 \sqrt{5} - 2 \sqrt{5}\)** can be combined because they have the same square root as a factor.

Step 2: Combine Like Terms with the Same Radical

Since **\(4 \sqrt{5}\)** and **\(-2 \sqrt{5}\)** are like terms (they both contain \(\sqrt{5}\)), we can subtract them:\(4 \sqrt{5} - 2 \sqrt{5} = 2 \sqrt{5}\).The result is \(2 \sqrt{5}\).

Step 3: Write the Final Expression

After combining the like terms, the expression simplifies to **\(3 \sqrt[3]{5} + 2 \sqrt{5}\)**. Since cubic roots and square roots are not like terms, this is the simplest form of the expression.

Key Concepts

Like Terms

In mathematics, the concept of "like terms" is very important when working with expressions. Like terms are terms that have the same variables raised to the same power, which allows them to be combined easily. A simple way to identify them is to look for terms that have the same literal component, or in the context of radicals, the same type of root or base.

For example:

• Terms like \(4\sqrt{5}\) and \(-2\sqrt{5}\) are like terms because both have the square root of 5, which means they can be combined by adding or subtracting their coefficients.
• However, \(3\sqrt[3]{5}\) is not like the terms \(4\sqrt{5}\) or \(2\sqrt{5}\), as it consists of a cubic root rather than a square root. Therefore, it stands alone in the expression.

Understanding like terms helps simplify expressions by effectively combining them, which often makes solving equations or further operations more manageable.

Cubic Roots

Cubic roots are a type of radical and involve finding a number which, when multiplied by itself three times, gives the original number. The notation for cubic roots is \( \sqrt[3]{x} \), where \(x\) is the number under the radical sign.

Some key points:

• The cubic root of a number can be positive or negative, depending on the sign of the number. For example, \( \sqrt[3]{8} = 2 \) and \( \sqrt[3]{-8} = -2 \).
• Unlike square roots, which always result in non-negative numbers, cubic roots can give negative outputs because multiplying a negative number three times results in a negative number.

Understanding cubic roots helps in algebra when solving equations that involve variables raised to the third power, as they are essential in reversing such powers. It is crucial to distinguish cubic roots from other roots because they follow different multiplication rules.

Square Roots

Square roots are another very common type of radical expression, and they represent a number which, when multiplied by itself, yields the original number. The notation for square roots is \( \sqrt{x} \).

Important aspects to remember include:

• Square roots of positive real numbers are always non-negative. For instance, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
• Square roots are not considered like terms with other roots, such as cubic roots, because their multiplication forms differ.

The ability to work with square roots is fundamental in many areas of algebra, helping to simplify expressions, solve quadratic equations, and perform operations on real numbers effectively. Being comfortable with recognizing and handling square roots is essential for progressing in mathematics.