Problem 43 - College Algebra with Modeling and Visualization

Question

Write the expression in radical notation. $$ (\mathrm{xy})^{1 / 2} $$

Step-by-Step Solution

Answer

\(\sqrt{\text{xy}}\)

Step 1: Understanding the Exponent

The expression given is \(( ext{xy})^{1/2}\). Here, the exponent \(1/2\) indicates that we need to express this using radical notation.

Step 2: Applying Radical Notation

In radical notation, an exponent of \(1/2\) means that we are taking the square root of the base. Therefore, the expression \(( ext{xy})^{1/2}\) can be rewritten as \(\sqrt{\text{xy}}\).

Key Concepts

Square Root

The square root is a concept that emerges frequently when dealing with radical notation. Essentially, finding the square root of a number or an expression means identifying a value that, when multiplied by itself, produces the original number. For example, if you have \(9\), its square root is \(3\), since \(3 \times 3 = 9\).
\(\) In mathematical terms, the square root is symbolized by the radical sign \(\sqrt{}\), so when we say \(\sqrt{9}\), we mean the number \(3\).
\(\) This concept becomes crucial in algebra to solve equations and simplify expressions, especially those involving exponents.

Exponents

Exponents are a fundamental part of algebra, representing how many times a number, known as the base, is multiplied by itself. An exponent is displayed as a small number to the upper right of the base number. For instance, in \(2^3\), the base is \(2\), and the exponent is \(3\), which means \(2 \times 2 \times 2 = 8\).
\(\) When dealing with exponents like \(\frac{1}{2}\), it signifies that we are finding the number that, when squared, gives us the original base number. This exponent specifically relates to square roots in that raising a number to the power of \(\frac{1}{2}\) is equivalent to taking the square root. So, \(\text{xy}^{1/2}\) becomes \(\sqrt{\text{xy}}\). This relationship makes exponents extremely useful in converting expressions between exponential and radical forms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and at least one arithmetic operation, such as \(+, - , \times, \div\). These expressions are the building blocks of algebra. They allow math to represent real-world problems in a generalized form.
\(\) Consider the expression \((\text{xy})^{1/2}\), which includes a variable component and an exponent. This flexibility means that algebraic expressions can be manipulated algebraically through operations like addition, subtraction, factoring, and expansion.

Working with algebraic expressions involves understanding how to simplify them using rules of arithmetic, exponents, and roots. Recognizing forms such as \(\sqrt{}\) helps to transition between different algebraic forms, making it easier to solve equations.