Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Fractional and Demical Exponents. $$(2.85)^{2 / 3}$$

Short answer

The value of \(2.85^{2 / 3}\) rounded to three significant figures is approximately 2.30.

Step-by-step solution

Understand Significant Figures

Before proceeding with the calculation, it's important to notice the number of significant figures in the given number. The number 2.85 has three significant figures because all non-zero digits are considered significant.

Apply the Exponent

To calculate \(2.85^{2/3}\), you'll have to find the cube root of 2.85 and then square that result. The cube root is equivalent to raising to the power of 1/3.

Calculate the Cube Root

Calculate the cube root of 2.85, which is \(2.85^{1/3}\). In most calculators, this can be done by typing in 2.85 and then using the cube root function or raising the number to the power of \(1/3\).

Square the Result

After finding the cube root, square that value to get \(2.85^{2/3}\). On a calculator, simply square the result of the previous step.

Retain Significant Digits

After squaring the cube root, make sure the final answer retains the correct number of significant figures, which is three in this case.

Key concepts

Fractional Exponents

Fractional exponents represent powers that involve both a root and an exponent. In the expression (2.85)^{2/3}, the denominator of the exponent, 3, signifies the root being taken—in this case, the cube root. The numerator, 2, tells us that once we've found this root, it should be squared.

To navigate such expressions, it's helpful to break them down step by step. First, find the cube root of the base, which gives us (2.85)^{1/3}. Next, raise this result to the power of 2. It's a two-step process that involves root extraction followed by exponentiation.

Decimal Exponents

When exponents are expressed in decimal form, they might at first seem less intuitive. However, decimal exponents can be converted into fractional form to simplify calculations. For instance, 2.85^{0.6667} (approximation of 2/3) is equivalent to 2.85^{2/3}. Calculators usually compute decimal exponents without needing conversion, but understanding the relationship to fractional exponents enhances conceptual knowledge.

Practical Calculation Tip

When typing into a calculator, it's more precise to use the exact fractional exponent rather than a decimal approximation, as rounding can lead to a minor loss of accuracy.

Significant Digits

The number of significant digits—often called significant figures or 'sig figs'—in a measurement or calculation indicates the precision of that value. It's a critical aspect in scientific and technical fields because it affects the certainty of the reported results.

To count significant digits, start from the first non-zero digit and count all digits until the end of the number, including any zeros that are sandwiched between non-zero digits or following a decimal point. The number 2.85 has three significant digits, meaning our final answer to any calculation involving this number should also be reported with three significant digits to maintain the precision of our initial value.

Cube Root Calculation

Calculating the cube root of a number involves finding a value that, when multiplied by itself three times (cubed), gives the original number. In our example with 2.85, the cube root is the number that when cubed equals 2.85.

Most modern calculators come equipped with a cube root function. However, you can also calculate this by raising a number to the power of 1/3. In our example, we calculate 2.85^{1/3} to find the cube root. It's important to retain the number of significant digits from the original number in our final cube root result.