Problem 56 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Fractional and Demical Exponents. $$(2.48)^{1.42}$$

Step-by-Step Solution

Answer

The result of (2.48)^{1.42} rounded to three significant digits is approximately 3.46.

Step 1: Understanding the Problem

We are asked to evaluate the expression with a decimal base of 2.48 raised to a decimal exponent of 1.42. When dealing with decimal exponents, it indicates a power and a root. A calculator is typically required for non-integer exponents, and the answer must be in proper significant digits.

Step 2: Calculating the Expression

To calculate the expression, enter the base number (2.48) into the calculator followed by the exponentiation function, then enter the exponent (1.42). The calculator will give a numerical approximation of the result.

Step 3: Retaining Significant Digits

Since our base of 2.48 has three significant digits, our final answer should also have three significant digits to retain the proper precision. Round the result from the previous step to three significant digits.

Key Concepts

Significant Digits

When working with measurements and calculations in science and mathematics, it's critical to express the results with the correct precision. This precision is conveyed through the use of significant digits, also known as significant figures. These are the digits in a number that carry meaning contributing to its precision. This includes all numbers other than:

• Zeros which serve merely as placeholders (they do not hold value unless they come after a decimal point and a non-zero number).
• Any additional digits that are not measured directly but estimated.

When performing calculations like exponentiation, as in the exercise \(2.48)^{1.42}\), it is essential to maintain the number of significant digits to reflect the certainty of the original data. After finding the approximated numerical value using a calculator, we must round the result to the same number of significant digits as the least certain figure used in our calculation, in this example, three significant digits from the base number 2.48.

Exponentiation

The concept of exponentiation extends beyond whole numbers to include fractions and decimals, allowing for a broader range of transformations. An exponent, like 1.42 in \(2.48)^{1.42}\), signifies repeated multiplication of the base, 2.48 in this case. However, when the exponent is not a whole number, it refers to both powers and roots. For instance, an exponent of 0.5 would represent a square root.

Exponentiation with decimal exponents follows the same principles as with integers but often requires more complex calculations. These are typically performed with the assistance of calculators, and they require understanding of how these fractional powers relate to roots. For example, the exponent 1.42 can be thought of as raising 2.48 to the 142nd power and then taking the 100th root of the result.

Numerical Approximation

Many mathematical problems involving decimal exponents cannot be solved exactly and must instead be approached through numerical approximation. This technique estimates the value of a number or expression when a precise value is either unnecessary or impossible to obtain.

For the given problem, a calculator can be used to approximate \(2.48)^{1.42}\) since it is non-intuitive and difficult to compute by hand. Calculators use advanced algorithms to approximate the value of such expressions to a high degree of accuracy. The key is to understand that the result given by a calculator is only an estimate and should be rounded according to the relevant significant digits to ensure that it does not imply a false sense of precision beyond the original data.