Problem 18 - Brief 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: Understanding the Expression

We need to evaluate the expression \( 36^{1/2} \). The notation \( a^{1/2} \) refers to the square root of \( a \). This means we are looking for the number that, when multiplied by itself, gives \( 36 \).

Step 2: Finding the Square Root

Identify which number, when squared, equals 36. Considering small whole numbers: \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), \( 4^2 = 16 \), \( 5^2 = 25 \), and \( 6^2 = 36 \). We find that \( 6 \times 6 = 36 \), so \( 6 \) is the number we seek.

Step 3: Confirm the Solution

To ensure our solution is correct, recall that the square root operation is the reverse of squaring a number. Since \( 6^2 = 36 \), we confirm that \( 36^{1/2} = 6 \).

Key Concepts

Exponentiation

Exponentiation is a key mathematical operation that involves rising a base to the power of an exponent. This function is written as \( a^n \), where \( a \) is the base and \( n \) is the exponent. The result is the base multiplied by itself \( n \) times. If \( n \) is 2, it is often referred to as "squaring" the base.

• Example: \( 3^2 = 3 \times 3 = 9 \)
• Example: \( 5^3 = 5 \times 5 \times 5 = 125 \)

Exponentiation extends beyond whole numbers. When the exponent is a fraction, for instance, \( a^{1/2} \), it provides us with the concept of roots. This is because fractional exponents signify inverse operations. Specifically, raising a number to the power of \( 1/2 \) yields its square root. This can be tricky to understand initially but is simply another way to denote roots using exponents.

Square Root

The square root of a number is the value that, when multiplied by itself, results in the original number. It is an inverse operation to squaring a number. The square root is commonly expressed using a radical symbol \( \sqrt{} \), but it can also be represented using fractional exponents, such as \( 36^{1/2} \).

• Understanding \( \sqrt{36} = 6 \) involves recognizing that \( 6 \times 6 = 36 \).
• Similarly, \( 81^{1/2} = 9 \) because \( 9 \times 9 = 81 \).

Finding square roots typically involves intuition and testing small numbers until you find a match, especially when starting out. As you get more comfortable, you'll recognize square numbers more easily. It’s crucial to confirm your understanding by checking your work: if you found \( \sqrt{36} = 6 \), then check that \( 6^2 = 36 \) to ensure correctness.

Mathematical Notation

Mathematical notation is a system that uses symbols to represent numbers, operations, and other mathematical concepts. It allows mathematicians and students to express mathematical ideas clearly and concisely. Consider \( a \), the base, and \( n \), the exponent, in the expression \( a^n \).This system includes:

• Operators like \( +, -, \times, \div \). These indicate addition, subtraction, multiplication, and division.
• Powers and roots, noted as \( a^n \) and \( \sqrt{a} \), which represent exponentiation and radicands.
• Fractional notation such as \( \frac{1}{2} \), which can also describe operations like roots.

With such notations, concepts such as \( 36^{1/2} \) become clearer, linking fractional exponents to practical operations. Understanding how to interpret and use mathematical notation is fundamental to solving problems and communicating ideas effectively.