Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Exponent. $$(-1.85)^{-3}$$

Short answer

\((-1.85)^{-3} \approx -0.15873\) with five significant digits.

Step-by-step solution

Understand Negative Exponents

A negative exponent indicates that you must take the reciprocal of the base and then raise it to the absolute value of the exponent. For any non-zero number 'a' and any integer 'n', the rule is: \(a^{-n} = \frac{1}{a^{n}}\).

Apply the Negative Exponent Rule

Apply the negative exponent rule to \((-1.85)^{-3}\) by taking the reciprocal of -1.85 and raising it to the 3rd power: \((-1.85)^{-3} = \left(\frac{1}{-1.85}\right)^3\).

Calculate the Reciprocal

Calculate the reciprocal of -1.85 which is \(-\frac{1}{1.85}\).

Raise the Reciprocal to the Power of 3

Raise the reciprocal to the power of 3: \(\left(-\frac{1}{1.85}\right)^3 = -\left(\frac{1}{1.85^3}\right)\).

Evaluate the Expression

Evaluate the expression with proper significant digits: \(-\frac{1}{1.85^3} = -\frac{1}{6.299375}\) which approximately equals -0.15873 when rounded to five significant digits, as the original number had three significant digits.

Key concepts

Significant Digits

Understanding significant digits is crucial for accurately reporting the precision of a numerical value. Significant digits, also known as significant figures, are the digits in a number that carry meaning contributing to its measurement resolution. This includes all non-zero digits, any zeroes between non-zero digits, and any trailing zeros in a decimal fraction.

When performing operations like multiplication or division, the number of significant digits in the final answer should be the same as the number in the least precise measurement. For example, when evaluating expressions with negative exponents, the result should retain the same number of significant digits as the original number. In the given problem \( (-1.85)^{-3} \) with \( -1.85 \) having three significant digits, the evaluated result should also be rounded to three significant digits.

Reciprocal of a Number

The reciprocal of a number is simply the value that, when multiplied by the original number, results in 1. It is essentially a flip of the numerator and denominator in a fraction. For any non-zero number \( a \), its reciprocal is \( \frac{1}{a} \).

To find the reciprocal of a negative number, such as \( -1.85 \), the reciprocal is \( -\frac{1}{1.85} \), which retains the negative sign. The concept of reciprocals comes into play when dealing with negative exponents, as this operation is fundamentally about finding the reciprocal of the base and then raising it to the positive exponent.

Evaluating Expressions

The process of evaluating expressions involves applying mathematical operations to simplify or find the value of expressions. In the context of expressions with negative exponents, evaluating them requires understanding exponent rules and operations with significant digits.

To successfully evaluate an expression like \( (-1.85)^{-3} \), you would first identify the negative exponent, finding the reciprocal of the base, and then raising that reciprocal to the power specified by the positive value of the exponent. The final answer is determined by paying close attention to the rules of significant digits to ensure the correct level of precision.

Exponent Rules

A firm grasp of exponent rules is essential for simplifying and evaluating mathematical expressions with exponents. One of these important rules is that for any non-zero number \( a \) and any integer \( n \) that \( a^{-n} = \frac{1}{a^{n}} \), which states that a negative exponent denotes that you should take the reciprocal of the base and raise it to the absolute value of the exponent.

In the context of evaluating \( (-1.85)^{-3} \), applying the negative exponent rule requires inverting -1.85 to find its reciprocal and then cubing this reciprocal. It is also crucial to remember that performing operations like raising to the power should be done with attention to the number of significant digits to ensure precision is not lost in the final answer.