Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Exponent. $$(1.83)^{-5}$$

Short answer

\((1.83)^{-5} = \frac{1}{(1.83)^5}\), with the final result rounded to three significant digits.

Step-by-step solution

Understanding Negative Exponents

A negative exponent indicates that the base should be inverted (reciprocal taken) and then raised to the corresponding positive exponent. For an expression of the form \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\).

Rewrite the Expression

The given expression \((1.83)^{-5}\) can be rewritten by taking the reciprocal of the base and changing the exponent to positive: \(\frac{1}{(1.83)^5}\).

Calculating the Positive Exponent

By performing the calculation \((1.83)^5\), we determine the value of the base raised to the positive exponent. This forms the denominator of the fraction.

Retain Significant Digits

Since our original base number has three significant digits (1.83), the final result must also reflect three significant digits, respecting the rules for significant figures in the calculation.

Final Calculation

After calculating the value of \((1.83)^5\), take its reciprocal to find the final answer. The answer should be rounded or formatted to maintain three significant digits.

Key concepts

Significant Digits

When dealing with numerical calculations, especially in science and engineering, the precision of the numbers used is dictated by significant digits (also referred to as significant figures). These are the digits in a decimal number that carry meaning towards its precision. The rules for determining significant digits include:

• Non-zero digits are always significant.
• Any zeros between non-zero digits are significant.
• Leading zeros are not significant because they serve only as placeholders.
• Trailing zeros in a decimal number are significant because they imply precision.

In calculations, the number of significant digits in the final result should match the least number of significant digits in any of the original numbers. So if you multiply two numbers, each with three significant digits, your result should also be rounded to three significant digits. This is crucial to consider when simplifying expressions with negative exponents because the conversion might affect how many significant digits your final answer has.

Reciprocal of a Number

The reciprocal of a number is essentially a flip of that number if it's written in fraction form. In other words, the reciprocal of a given number 'a' is 1 divided by 'a' (\(\frac{1}{a}\) when 'a' is not zero). For any non-zero number, the product of the number and its reciprocal is always 1. This is an important property that comes into play with negative exponents. For instance, the reciprocal of 1.83 is \(\frac{1}{1.83}\), and when using negative exponents, you convert the base to its reciprocal and change the negative exponent to a positive one. Understanding reciprocals is vital for handling divisions and simplifying fractions in algebra.

Exponentiation

Exponentiation is a mathematical operation where a number (called the base) is multiplied by itself a certain number of times specified by the exponent. If the exponent is negative (--5 in this case), the base is replaced by its reciprocal, and the exponent sign is changed to positive. The general form looks like \(a^n\) where 'n' is the number of times 'a' is used as a factor in the multiplication. For negative exponents, the expression \(a^{-n}\) becomes \(\frac{1}{a^n}\). It's key to remember that the base itself does not change, only its position with respect to the fraction bar changes due to exponentiation rules. In our example with the base 1.83 and a negative five exponent, you are actually calculating the reciprocal of \(1.83^5\), not changing 1.83 in any other way.