Question

Use a calculator to evaluate the expression. Round your answer to two decimal places. $$x^{3}-8.29 \text { when } x=-2.47$$

Short answer

After performing the calculations, rounding off the result to two decimal places will yield the final answer.

Step-by-step solution

Substitute x value

In the first step, substitute the value of \(x=-2.47\) into the expression \(x^{3}-8.29\). It will lead to the following: \((-2.47)^{3}-8.29\)

Calculate the Expression

Calculate \((-2.47)^{3}-8.29\). Multiply -2.47 with itself three times and then subtract 8.29 from the result obtained.

Round the answer to 2 decimal places

As a final step, round off the result to two decimal places. If the third digit after the decimal point is equal to or greater than 5, increase the second digit by one. If it's less than 5, keep the second digit unchanged.

Key concepts

Substitution in Algebra

Substitution is a fundamental concept in algebra which involves replacing a variable with its corresponding numerical value. In the case of evaluating an algebraic expression using a calculator, substitution is the first critical step. For example, if you have an expression such as \(x^{3} - 8.29\) and you are asked to find its value for \(x = -2.47\), you start by replacing every occurrence of \(x\) in the expression with -2.47. This process changes the algebraic expression into a numerical expression that you can calculate. It's essential to follow the order of operations while performing calculations to ensure accuracy.

Correctly performing substitution allows for an accurate evaluation of expressions, especially those involving exponents and other operations. Always remember to perform the substitution step-by-step to avoid errors. This means replacing each instance of the variable with the given number before performing any calculations.

Exponents and Powers

Dealing with exponents and powers is a key component of working with algebraic expressions. Exponents like the one in \(x^{3}\) indicate how many times to multiply the base (in this case, \(x\)) by itself. It's a shorthand for repeated multiplication: \(x^{3}\) is equivalent to \(x \times x \times x\). When you encounter a negative base raised to an exponent, like \(-2.47^{3}\), you may find that the result is negative because an odd power of a negative number remains negative.

When calculating exponents on a calculator, be mindful of the placement of parentheses, as they can affect the outcome. Without parentheses, some calculators will interpret \(-2.47^{3}\) incorrectly as \(-1 \times 2.47^{3}\), thus it's crucial to enter it as \((-2.47)^{3}\) to get the accurate result. Understanding exponents and powers thoroughly aids in simplifying and evaluating algebraic expressions effectively.

Rounding Decimals

Rounding decimals is a means to simplify numbers and tailor precision in calculations. In many mathematical problems, you may be instructed to round your final answer to a certain number of decimal places. For instance, if an answer must be rounded to two decimal places, you look at the third decimal place. If this digit is 5 or more, you increase the second decimal place by one, effectively rounding up. If the digit is less than 5, you keep the second decimal place unchanged, rounding down.

Applying this to our original problem, if the calculator gives you an answer like -19.221 for \((-2.47)^{3} - 8.29\), you would observe the third decimal place. Since the digit '1' is less than 5, the rounded number to two decimal places will be -19.22. Rounding is a simple yet important aspect of numerical expressions, especially in fields that demand specific levels of precision like engineering, physics, and finance. Remember, the goal of rounding is to provide a number that is as close as possible to the original number while maintaining the requested level of precision.