Problem 36 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(\sqrt{3xy}\)

Step 1: Understand the Expression

The given expression is \((3xy)^{\frac{1}{2}}\). The goal is to convert this expression from exponential form to radical form.

Step 2: Identify the Root

The exponent \(\frac{1}{2}\) indicates that we need to take the square root of the base. In radicals, the expression \(b^{\frac{1}{n}}\) translates to the \(n\)-th root of \(b\). Therefore, \(\frac{1}{2}\) implies a square root.

Step 3: Express in Radical Form

Convert \((3xy)^{\frac{1}{2}}\) into radical form. The expression becomes \(\sqrt{3xy}\), representing the square root of \(3xy\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication. When you see an expression like \(a^n\), it means that the base \(a\) is multiplied by itself \(n\) times. Exponents simplify the way we express large numbers and perform calculations. For example, \(4^3\) is equivalent to multiplying 4 by itself three times: \(4 \times 4 \times 4 = 64\).
When dealing with fractional exponents like \(b^{\frac{1}{n}}\), the numerator of the fraction keeps indicating power, and the denominator indicates the type of root. For instance, \(x^{\frac{1}{2}}\) represents the square root of \(x\), while \(x^{\frac{1}{3}}\) represents the cube root of \(x\).

• A fractional exponent of \( \frac{1}{n} \) turns into the \(n\)-th root.
• If the fraction is \( \frac{m}{n} \), it means \(x^m\) and then take the \(n\)-th root of it.

Square Root

The square root is a type of radical symbolized by \( \sqrt{} \). Finding the square root is essentially determining which number multiplied by itself results in the given number. For example, \( \sqrt{16} = 4 \) since \(4 \times 4 = 16\).
The term inside the square root symbol is called the radicand, and the root of the radicand must be a non-negative number because the square of a negative number will be positive as well. This principle is crucial especially when working with real numbers.

• The square root of \(x\) is represented as \(x^{\frac{1}{2}}\).
• Thus, \((3xy)^{\frac{1}{2}}\) can be rewritten as \(\sqrt{3xy}\).

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. For example, in an expression like \(3xy\), 3 is a coefficient, while \(x\) and \(y\) are variables. These expressions allow us to describe real-world situations abstractly and solve for unknowns.
When evaluating or transforming algebraic expressions, understanding how to manipulate them through operations such as addition, subtraction, multiplication, division, and exponentiation is crucial.

• Converting expressions into different forms, such as from exponential to radical, helps in simplifying and solving equations.
• An example is turning \((3xy)^{\frac{1}{2}}\) into \(\sqrt{3xy}\), which can make further calculations more straightforward.