Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Fractional and Demical Exponents. $$(8.88)^{2.13}$$

Short answer

The value of \((8.88)^{2.13}\) rounded to three significant digits is approximately 76.5.

Step-by-step solution

Understanding Significant Digits

Before performing the calculation, it is important to identify the number of significant digits in the given number. The number 8.88 has three significant digits, as all non-zero digits and zeros between them are considered significant.

Performing the Calculation

To calculate the value of the expression, raise 8.88 to the power of 2.13 using a calculator or any other computational tool that can handle powers.

Rounding to Proper Number of Significant Digits

Once the calculation is done, round the answer to three significant digits because the number with the least significant digits in the expression is 8.88, which has three significant digits.

Key concepts

Fractional Exponents

Fractional exponents are a way of expressing powers and roots combined and denote the operation of raising a number to a rational power. For instance, the expression \( a^{\frac{b}{c}} \) can be interpreted as the c-th root of \( a \) raised to the power of b. An important thing to remember is that when dealing with fractional exponents in a calculation, the rules for significant digits still apply.

The process of evaluating fractional exponents involves transforming them into a form that is easier to comprehend and compute. For instance, \( 8^{\frac{3}{2}} \) can also be written as \( \sqrt[2]{8^3} \) or \( (\sqrt[2]{8})^3 \), which calculates to 4 raised to the third power or 64. Significant digits come into play when dealing with the precision of these calculated values and are crucial for maintaining accuracy in mathematical and scientific contexts.

Decimal Exponents

Decimal exponents, just like fractional exponents, also represent the power to which a number is raised, but instead of a fraction, a decimal is used. The base number raised to a decimal exponent involves taking roots as in fractional exponents, in addition to powers. When using decimal exponents such as in the expression \( 8.88^{2.13} \), it's important to note that the exponent itself does not affect the number of significant digits within the base number.

Calculating a decimal exponent may require a scientific calculator or software that can handle such computations. One way to work with decimal exponents is to convert them into fractional exponents, as some calculators can handle fractions more readily. Upon solving, the result must be rounded according to the rules of significant digits to ensure that the final answer is reported with the appropriate level of precision.

Rounding to Significant Digits

Rounding to significant digits ensures that numerical answers reflect the precision implied by the given data. It is a critical part of reporting numerical results in mathematics and the sciences. In an expression like \( (8.88)^{2.13} \), once the calculation is performed, the result should be rounded to match the least number of significant digits from any of the numbers involved in the computation.

To properly round a number, you should:

• Locate the last significant digit that should be retained.
• Look at the next digit (to the right) to determine if you should round up.
• If the next digit is 5 or more, you round up the last significant digit. If it is less than 5, you leave it as is.

For example, if the solution to an expression is 78.945 and you need to round it to three significant digits, the answer would be 78.9. Rounding reflects the precision of the input and is a best practice when expressing numerical solutions.