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Chapter 1: Problem 51

Use a calculator to evaluate each expression. Round answers to two decimal places. $$ 8^{2.7} $$

Short Answer

Expert verified
The evaluated expression \( 8^{2.7} \) rounded to two decimal places is approximately 512.86.

Step by step solution

01

Understand the Expression

The expression given is an exponential expression: \( 8^{2.7} \). This means we need to calculate the value of 8 raised to the power of 2.7.
02

Setup for Calculation

We will use a calculator to find the value of \( 8^{2.7} \). Ensure your calculator is set to handle exponential calculations.
03

Input the Expression into the Calculator

On your calculator, enter the base number, which is 8. Then use the power or exponential function (usually shown as '^' or '**'), and input the exponent, which is 2.7.
04

Compute with the Calculator

After entering \( 8^{2.7} \) into the calculator, press the equals button to compute the result.
05

Round the Result

The calculator will give you a result. Round this value to two decimal places, as requested in the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculator Usage
When you encounter a problem requiring exponential calculations like evaluating \( 8^{2.7} \), using a calculator can simplify your work significantly. Modern calculators, especially scientific ones, come equipped with a power function, often signified by symbols like '^' or '**'. These functions allow you to efficiently calculate the powers of numbers without manual computation.

Here鈥檚 a simple guide on using a calculator for such calculations:
  • Enter the base of the power, which is the larger number being multiplied, such as 8 in this scenario.
  • Input the exponent, which indicates how many times the base number is used in repeated multiplication. For this task, that's 2.7.
  • Press the equals button to execute the calculation and retrieve the result.
This method saves you from tedious arithmetic and improves accuracy. Make sure that your calculator is set to handle such functions correctly before starting.
Rounding Numbers
Rounding numbers is a crucial skill, especially when dealing with precise calculations and numerical analysis. Once you obtain a result from your calculator, it might be a long decimal number. However, many problems, like the one we're solving, ask you to round numbers to a specific number of decimal places. In our case, the result from \( 8^{2.7} \) needs to be rounded to two decimal places.

When rounding numbers:
  • Focus on the digits after the decimal point. For rounding to two decimal places, observe the third decimal place.
  • If the third decimal place is 5 or higher, increase the second decimal place by one.
  • If it's less than 5, simply truncate the number after the second decimal place.
Rounding helps in reducing complexity while providing a more practical and comprehensible number for further interpretations or computations.
Exponential Functions
Exponential functions are fundamental in mathematics and appear frequently in various fields including finance, physics, and biology. An exponential expression like \( 8^{2.7} \) involves a base number (8) and an exponent (2.7). This denotes the number being multiplied by itself a specific number of times, often resulting in a very large or very small number.

Key traits of exponential functions include:
  • Growth or decay that is proportionate to the current value, making them distinct from linear patterns.
  • The base of the exponent signifies the initial quantity being repeatedly multiplied.
  • The exponent represents the power to which the base is raised, controlling how many times multiplication occurs.
For bases greater than 1, the function grows as the exponent increases, known as exponential growth. Conversely, when the base is between 0 and 1, there is exponential decay. Understanding these characteristics is vital for working competently with exponential functions.

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Most popular questions from this chapter

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