Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Negative Exponent. $$(3.84)^{-3}$$

Short answer

0.0176

Step-by-step solution

Understanding Negative Exponents

A negative exponent means that the number is the reciprocal of the number with a positive exponent. In other words, to evaluate an expression like \((a)^{-n}\), you would calculate \((1/a)^{n}\).

Apply the Negative Exponent Rule

For the expression \((3.84)^{-3}\), apply the negative exponent rule by changing it to \((1/3.84)^{3}\).

Calculate the Reciprocal

First, calculate the reciprocal of 3.84, which is \(1/3.84 = 0.26041667\).

Raise the Reciprocal to the Positive Power

Next, raise the reciprocal to the third power: \((0.26041667)^{3}\).

Compute the Power

Calculate the value of \((0.26041667)^{3}\) using a calculator or long multiplication to get approximately 0.01757812.

Round to the Correct Number of Significant Digits

Since the original number 3.84 has three significant digits, the final answer should be rounded to three significant digits as well. The result is 0.0176.

Key concepts

Significant Digits

Significant digits, also called significant figures, are a vital concept in precision measurement. They refer to all the non-zero digits in a number, plus any zeroes sandwiched between them or trailing after a decimal point and a non-zero digit. For instance, in the number 123.00, all five digits are significant because the zeroes indicate precision.

When performing calculations, it's essential to retain the proper number of significant digits to reflect the true precision of your result. If we start with a number that has three significant digits, like 3.84, any mathematical operations we perform should yield a result also rounded to three significant digits. This ensures we don't overstate the precision of our answer and remain consistent with our initial data's accuracy.

Reciprocal of a Number

The reciprocal of a number is simply one divided by that number. It's also known as the multiplicative inverse, because when you multiply a number by its reciprocal, you get one. To find the reciprocal of a number, like 3.84, you would calculate \(1/3.84\).

In terms of notations, the reciprocal of a number 'a' would be expressed as \(1/a\) or \(a^{-1}\). This concept is especially prominent when dealing with negative exponents, as they indicate that we should use the reciprocal of the base number raised to the opposite positive exponent. Reciprocals are a fundamental building block in algebra and play a crucial role in solving equations that involve division.

Exponentiation

Exponentiation is an arithmetic operation where a number, known as the base, is multiplied by itself a certain number of times, indicated by the exponent. For example, \(3^4\) means multiplying 3 by itself 4 times: \(3 \times 3 \times 3 \times 3\).

When the exponent is negative, it means that we take the reciprocal of the base and then raise it to the corresponding positive exponent, as we've seen with the negative exponent rule. So \(a^{-n}\) becomes \(1/a^n\). Exponentiation is a powerful tool that not only appears in basic arithmetic but also in more complex fields like calculus, statistics, and financial modeling.