Problem 3 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Powers. $$9^{2}$$

Step-by-Step Solution

Answer

81 with all significant digits retained.

Step 1: Understanding powers

When raising a number to a power, this means you multiply the number by itself as many times as indicated by the exponent. Here, we have 9 raised to the power of 2, denoted as 9^2, which means 9 is multiplied by itself once.

Step 2: Calculating the power

To evaluate 9^2, multiply 9 by itself: 9 * 9 = 81.

Step 3: Retaining significant digits

The number 9 has one significant digit, but since it's an exact count (a square of a whole number), it retains all significant digits after the calculation. Therefore, the answer, 81, retains both digits as significant.

Key Concepts

Evaluating Powers

Evaluating powers, such as calculating the result of an expression like \(9^2\), is a fundamental operation in mathematics. To evaluate an expression with an exponent, you multiply the base, in this case, the number 9, by itself as many times as the exponent indicates. For \(9^2\), this implies a single multiplication of 9 by itself.

It is essential to understand that when a number is raised to the second power, often referred to as squaring, the resulting value is always positive. This is because multiplying two positive or two negative numbers always yields a positive result. For example, \((-9)^2\) is also \(81\), just as \(9^2\) is. This concept is not restricted to positive integers. Decimals and fractions can also be squared by using the same multiplication principle but require careful attention to maintain accuracy and the correct number of significant digits in the result.

Multiplication of Numbers

Multiplication of numbers is one of the four basic arithmetic operations, allowing us to compute the total of equal groups of objects. When multiplying two numbers, such as during the evaluation of powers, the order in which you multiply the numbers does not affect the result due to the commutative property of multiplication. This means that \(9 \times 9\) is equivalent to \(9 \times 9\).

Furthermore, understanding multiplication of numbers is crucial because it serves as the basis for more complex operations like powers and plays a significant role in various fields, including finance, science, and engineering. Basic multiplication skills are also the cornerstone for learning other aspects of mathematics, such as division, fractions, and algebra.

Rules for Significant Digits

Significant digits are critical in scientific and mathematical calculations because they indicate the precision of measured or calculated quantities. The rules for significant digits (or significant figures) help determine which digits in a number are meaningful and contribute to its precision.

• In whole numbers without a decimal point, only the non-zero digits are considered significant. For example, in the number 450, there are two significant digits: 4 and 5.
• Any zeros between significant digits are also significant. Hence, in 405, the zero is significant.
• All non-zero digits in decimal numbers are significant. For example, 0.009 has one significant digit: 9.
• Zeros to the right of the decimal point and at the end of a number are significant if the number contains a decimal point. In 45.00, all four digits are significant.

Applying these rules to the initial problem of \(9^2\), you recognize that the base, 9, is a whole number and has one significant digit. After the calculation, the answer, 81, retains both of its digits as significant because it's an exact square of the integer 9. In other circumstances, when a calculation involves inexact numbers (such as measurements), care must be taken to ensure the final result reflects the correct number of significant figures based on these rules.