Problem 35 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Exponent. $$10^{-2}$$

Step-by-Step Solution

Answer

The value of \(10^{-2}\) is 0.01.

Step 1: Understanding Negative Exponents

A negative exponent indicates that we need to take the reciprocal of the base to the absolute value of the exponent. In this case, the base is 10, and the exponent is -2. The reciprocal of 10 is 1/10.

Step 2: Calculating the Value

To evaluate the expression, we write 10 to the power of -2 as 1 over 10 to the power of 2.

Step 3: Final Evaluation

We then calculate the power of 10. Since 10 squared is 100, we take the reciprocal, which is 1 over 100 or 0.01 in decimal form.

Key Concepts

Understanding Negative Exponents

When dealing with negative exponents, it's essential to recognize that they signify the reciprocal of the base raised to the positive value of the exponent. The expression with a negative exponent, like \(10^{-2}\), can be intimidating at first, but it's actually straightforward once you understand the process.

In solving \(10^{-2}\), we find the reciprocal of the base, which is 10 in this case. The reciprocal of a number is simply 1 divided by that number. Consequently, the reciprocal of 10 is \(\frac{1}{10}\). From there, we consider the positive exponent by simply raising the reciprocal to the absolute value of the negative exponent. Thus, \(10^{-2}\) becomes \(\left(\frac{1}{10}\right)^2\). This leads us to the final step, evaluating the expression by carrying out the exponentiation, which ultimately provides us with our answer.

Significant Digits

The concept of significant digits, often referred to as significant figures, is a fundamental part of dealing with numbers in science and mathematics, especially when it comes to precision. These are the digits in a number that carry meaning contributing to its precision. This includes all digits except:

• Leading zeros, which serve to position the decimal point.
• Trailing zeros when they are merely placeholders to indicate the scale of the number.
• Spurious digits that arise beyond the resolution of the measurement.

In our example, once we've calculated \(10^{-2}\) to be 0.01, we observe that there are two significant digits—1 and another 1, because the zeros before the 1 are not significant. Maintaining proper significant digits is crucial when reporting measurements or in precise calculations to ensure that the precision of the answer best reflects the accuracy of the given data.

Power of Ten

A 'power of ten' refers to the expression where ten is raised to a given exponent. When an exponent is positive, the power of ten will be a large number, indicating how many times to multiply the number 10 by itself. On the contrary, when the exponent is negative, it indicates the division instead of multiplication, demonstrating how many times to divide 1 by 10.

In our original exercise, \(10^{-2}\) is a power of ten with a negative exponent. Calculating the power of ten provides a systematic approach to understanding both large and small numbers, simplifying their representation and computation. It's especially helpful when working with scientific notation, where numbers are written as a product of a number between 1 and 10 and a power of ten, letting us express both very large and very small numbers efficiently. For instance, 0.01 can be written as \(10^{-2}\) in scientific notation.