Problem 17 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 25^{1 / 2} $$

Step-by-Step Solution

Answer

\( 25^{1/2} = 5 \).

Step 1: Understanding the Expression

The expression given is \( 25^{1/2} \). The notation \( a^{1/2} \) refers to the square root of \( a \). So, \( 25^{1/2} \) means the square root of 25.

Step 2: Identify the Square Root

We need to find the number which, when multiplied by itself, equals 25. The number that satisfies this condition is 5, since \( 5 \times 5 = 25 \).

Step 3: Finalize the Answer

Since the square root of 25 is 5, we conclude that \( 25^{1/2} = 5 \).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics that represent repeated multiplication. When we encounter a number expressed as a base with an exponent, like \( a^n \), it means we multiply the base \( a \) by itself \( n \) times. For example, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \). There are a few special cases to be aware of:

• Any number to the power of 1 is the number itself (e.g., \( 7^1 = 7 \)).
• Any number to the power of 0 is 1, except for 0 itself (e.g., \( 9^0 = 1 \)).

A fractional exponent, like \( a^{1/n} \), signifies the \( n \)-th root of \( a \). Therefore, \( 25^{1/2} \) is the square root of 25. Understanding these basics is crucial when working with expressions involving exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form so they are easy to work with or interpret. For expressions involving exponents, simplification can include applying rules of exponents and roots.

• When faced with a fractional exponent like \( x^{1/2} \), it indicates the square root, meaning you find the number that produces \( x \) when squared.
• For example, in \( 25^{1/2} \), simplify by identifying that 5 is the square root of 25, as \( 5 \times 5 = 25 \).
• Always check if you can further reduce or simplify. Sometimes expressions might already be in their simplest form.

By practicing these steps, you can become more efficient at working with expressions, making complex problems manageable.

Mathematical Notation

Mathematical notation is a language of symbols used to express concepts in mathematics succinctly and accurately. Clear understanding of this notation helps solve and communicate mathematical problems.

• For example, the expression \( 25^{1/2} \) uses the notation for exponents to indicate the square root of 25.
• Common symbols include:
  • \( +, -, \times, \div \) for basic operations.
  • \( ^ \) to denote exponents.
  • \( \sqrt{} \) to signify roots.
• Understanding these symbols allows you to interpret and solve problems systematically.

Mathematical notation streamlines the process of dealing with complex operations, and learning it is like learning a universal mathematical language.