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Chapter 0: Problem 17

Compute the numbers. $$8^{4 / 3}$$

Short Answer

Expert verified
The result is 16.

Step by step solution

01

Understand the Exponent

The exponent in the expression is a fraction, which means it involves both a root and a power. Specifically, the exponent \( \frac{4}{3} \) means taking the cube root of 8 first, and then raising the result to the power of 4.
02

Calculate the Cube Root

Find the cube root of 8. Recall that \( 2^3 = 8 \), so the cube root of 8 is 2.
03

Raise to the Power of 4

Now take the result from Step 2 (which is 2) and raise it to the power of 4. \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \).
04

Verify the Result

Double-check the calculations to ensure accuracy. The cube root of 8 is indeed 2, and \( 2^4 = 16 \).

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Key Concepts

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

cube root
Let's start by understanding what a cube root is. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, since \(2^3 = 8\), the cube root of 8 is 2.
Here's how you can imagine it:
  • Take the number 2.
  • Multiply it by itself 3 times: \(2 \times 2 \times 2\).
  • The result is 8.
So, to find the cube root of 8, you're looking for a number that meets this condition. Clearly, that number is 2.
Cubes and cube roots are essential concepts in algebra and are useful for simplifying expressions, solving equations, and understanding higher-level mathematics.
raising to a power
Raising a number to a power means multiplying the number by itself a certain number of times. The 'power' tells you how many times to use the number in the multiplication.
For example, in the expression \(2^4\):
  • The number 2 is the base.
  • The number 4 is the exponent or power.
  • We multiply 2 by itself four times: \(2 \times 2 \times 2 \times 2\).
Following these steps, you get 16 as a result.
Combining roots and exponents can sometimes be confusing, but breaking it down into smaller steps, like finding the cube root first and then raising it to a power, makes it much simpler.
verifying results
It's crucial to verify your results to ensure accuracy in your calculations. Even small mistakes can lead to wrong answers, so double-checking your work is always a good practice.
In our example, we confirmed that the cube root of 8 is 2 by identifying that \(2^3 = 8\). Next, we further checked that \(2^4 = 16\).
Here are some steps to verify your work systematically:
  • Retake the calculation step-by-step.
  • Ensure that each stage (root extraction and power raising) is correct independently.
  • Consider alternative methods or using a calculator for confirmation.
Developing the habit of verification helps solidify your understanding and increases your confidence in solving similar problems in the future.

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