Problem 18 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 36^{1 / 2} $$

Step-by-Step Solution

Answer

The expression evaluates to 6.

Step 1: Understand the Problem

The problem asks for the evaluation of the expression \(36^{1/2}\). This expression means we need to find the square root of 36.

Step 2: Recognize the Relationship

Recall that the notation \(a^{1/2}\) is equivalent to \(\sqrt{a}\). Thus, \(36^{1/2}\) can be rewritten as \(\sqrt{36}\).

Step 3: Find the Square Root

Identify the number that, when multiplied by itself, results in 36. Consider 6 because \(6 \times 6 = 36\). Thus, \(\sqrt{36} = 6\).

Key Concepts

Exponents and Radicals

Exponents and radicals are essential concepts in mathematics that help us simplify complex expressions. When we talk about exponents, we mean the number that indicates how many times to multiply a base number by itself. For example, in the number \(5^3\), 5 is the base and 3 is the exponent, signaling that 5 should be multiplied by itself three times, resulting in 125.

On the other hand, radicals are the opposite operation of exponents. They allow us to determine the root of a number. The most common radical is the square root, denoted by the symbol \(\sqrt{}\). A square root finds the original number that, when squared, returns the given number. Understanding these concepts fully ensures a grasp on many mathematical operations involving expressions like \(36^{1/2}\). Such expressions show the interconnection between exponents and radicals, effectively expressing the same operation in different formats.

Mathematical Notations

Math relies heavily on various notations to communicate different operations and expressions succinctly. Notation helps convert long and complex mathematical sentences into concise expressions. In the exercise, "36^{1/2}" is a great example of mathematical notation.

The 'caret' symbol (^) represents an exponentiation operation. Here, "1/2" as an exponent serves as a shorthand for the square root operation. By understanding this notation, students can easily interpret that \(36^{1/2}\) translates to \(\sqrt{36}\).

Recognizing these symbols is crucial, as many mathematical problems are simplified and streamlined into these formats. By mastering these notational representations, students can move more easily from one type of mathematical language to another, enhancing their ability to solve problems efficiently.

Arithmetic Operations

Arithmetic operations form the basis of much of mathematics and include addition, subtraction, multiplication, division, and taking roots. These operations provide the necessary tools to tackle more complex equations and expressions, such as evaluating \(36^{1/2}\).

For the expression \(36^{1/2}\), performing a square root operation is a key arithmetic task. Here, you identify the number which, multiplied by itself, gives you the original number (36).

• Think of it as finding a number "x" such that \(x \times x = 36\).
• In this case, "x" is 6, because \(6 \times 6 = 36\).

By understanding how to perform these basic arithmetic operations, students can break down complex processes into simpler, more manageable steps, leading to easier and more successful problem-solving in mathematics.