Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Fractional and Demical Exponents. $$(8.55)^{1 / 3}$$

Short answer

The cube root of 8.55, rounded to three significant digits, is approximately 2.04.

Step-by-step solution

Understand the Exponent Notation

When a number is raised to the fraction \(1/3\), it is equivalent to the cube root of the number. So \(8.55^{1/3}\) means the cube root of 8.55.

Calculating the Cube Root

Use a calculator or a cube root function to find the value of \(8.55^{1/3}\).

Retaining Significant Digits

The number 8.55 has three significant digits, so the answer should also be rounded to three significant digits. After calculating the cube root, round the result accordingly.

Key concepts

Fractional Exponents

When you come across a number raised to a fractional power, such as \(8.55^{1/3}\), understanding what this means is crucial. A fractional exponent indicates a root; specifically, the denominator tells you the type of root. In this case, the denominator is 3, which means we’re dealing with a cube root. If you had an exponent of \(1/2\), this would indicate a square root.

These fractional exponents follow the same rules as other exponents, but instead of the number being multiplied by itself a certain number of times, the number is being 'rooted'. For instance, \(x^{1/n}\) is the same as the nth root of x. Keeping this in mind will help you simplify and evaluate expressions with fractional exponents with ease.

Cube Root Calculation

To find the cube root of a number, you are looking for a value that, when multiplied by itself twice (which makes three times in total), gives you the original number. Calculating the cube root of 8.55 means we're in search of a number that, when cubed, equals 8.55.

You can often use a scientific calculator to directly compute cube roots or raise the number to the power of \(1/3\). Alternatively, some prefer using the ^ key or a specific cube root function. It's all about finding the method that works best for you. In classroom settings or standardized tests where a calculator might not be allowed, estimation and familiarity with perfect cubes can also help you deduce the cube root.

Significant Figure Rules

Working with significant figures is essential in accurately conveying the precision of a calculated value. The key rules to remember include:

• Non-zero digits are always significant.
• Any zeros between significant digits are significant.
• Leading zeros are not significant as they only indicate the position of the decimal.
• Trailing zeros in a number are significant if there is a decimal point present.

In the case of calculating the cube root of 8.55, we start with a number that has three significant digits. The result of our cube root calculation must also reflect three significant digits to maintain the same level of precision. Rounding, in this context, is not merely a matter of following mathematical convention; it's about ensuring that our answer remains as precise as the data we started with.