Problem 33 - Algebra for College Students

Question

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \(3 x^{\frac{1}{2}}\)

Step-by-Step Solution

Answer

The expression in radical form is \(3\sqrt{x}\).

Step 1: Identify the Components

The given expression is \(3x^{\frac{1}{2}}\). It consists of a numerical coefficient (3) and an algebraic term \(x^{\frac{1}{2}}\). Our goal is to express \(x^{\frac{1}{2}}\) in radical form.

Step 2: Convert the Exponent to Radical Form

The expression \(x^{\frac{1}{2}}\) signifies a radical form. The exponent \(\frac{1}{2}\) corresponds to the square root. Therefore, \(x^{\frac{1}{2}}\) can be written as \(\sqrt{x}\).

Step 3: Combine the Terms

Replace \(x^{\frac{1}{2}}\) with \(\sqrt{x}\) in the original expression. The entire expression becomes \(3\sqrt{x}\).

Key Concepts

Understanding Exponents

Exponents are a mathematical shorthand used to express repeated multiplication of a number by itself. They are represented as a small number, known as the "power," placed above and to the right of the base number. For instance, in the expression \(x^{n}\), \(x\) is the base and \(n\) is the exponent, indicating that \(x\) is multiplied by itself \(n\) times. If \(n\) is a fraction, like \(\frac{1}{2}\), it suggests taking a root of the base rather than multiplying it.

• Base: The number that is being multiplied or is under the power.
• Exponent: The power to which the base is raised, indicating how many times the base is used in the multiplication.

Exponents are pivotal in expressing large numbers succinctly and can help to convert expressions into more manageable forms by using roots. For example, \(x^{\frac{1}{2}}\) denotes the square root of \(x\), turning exponent-based calculations into root-based ones.

The Square Root Concept

A square root of a number is a value that, when multiplied by itself, gives the original number. It signifies finding an original number when given its square. The square root symbol is \(\sqrt{}\), which is essentially an inverse operation of squaring a number.

• Symbol: The most common symbol for a square root is \(\sqrt{}\).
• Calculation: To find the square root of 9, you find a number which when squared, results in 9: \(3 \times 3 = 9\), so \(\sqrt{9} = 3\).

Squares and square roots are useful in simplifying algebraic expressions and are evident in various equations and functions. In our given algebraic expression, turning an exponent into a square root provides an alternate way to interpret and solve mathematical problems.

Basics of Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition or multiplication). They represent mathematical phrases and allow the use of symbols to denote relationships and operations. Key elements include:

• Variables: Symbols, usually letters, that represent unknown values. For example, in \(3x^{\frac{1}{2}}\), \(x\) is a variable.
• Coefficients: Numerical values that multiply a variable, such as the 3 in the expression \(3x^{\frac{1}{2}}\).
• Terms: Parts of the expression, consisting of variables and coefficients. Each term is separated by addition or subtraction.

Algebraic expressions can take many forms, from simple integers to more complex polynomial expressions. They form the basis for equations, which are used to solve real-world and theoretical problems by setting expressions equal to each other. Understanding the makeup of these expressions is fundamental to manipulating and simplifying them, especially when converting expressions with exponents into radical forms.