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Chapter 4: Problem 84

Simplify. $$ 3^{4} $$

Short Answer

Expert verified
81

Step by step solution

01

Understand the Base and Exponent

In the expression \(3^4\), the base is 3, and the exponent is 4. This means you need to multiply the base (3) by itself repeatedly for a total of 4 times.
02

Multiply the Base for the First Two Factors

Start multiplying: \(3 \times 3 = 9\).
03

Multiply the Result by the Next Factor

Continue multiplying: Take the result from the previous step and multiply by the base again: \(9 \times 3 = 27\).
04

Multiply the Result by the Last Factor

Finish the multiplication: Take the result from Step 3 and multiply by the base one last time: \(27 \times 3 = 81\).

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Key Concepts

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

Base and Exponent
In the expression \(3^4\), we have two key components: the base and the exponent. The base here is 3, which is the number that will be repeatedly multiplied. The exponent is 4, indicating how many times the base will be used in the multiplication process.

Understanding this helps us know what the expression is asking us to do. Essentially, \(3^4\) is a shorthand way of saying "multiply 3 by itself a total of four times." This is the basic principle of exponentiation, where the base is the starting point and the exponent tells us the number of multiplications needed.
Multiplication
Once we've understood the idea of base and exponent, we move onto the multiplication process itself. Let's outline the steps for working through \(3^4\). This involves several layers of multiplication:
  • First layer: Start by multiplying the base by itself for the first two factors: \(3 \times 3 = 9\).
  • Second layer: Take the result (9) and multiply by the base again: \(9 \times 3 = 27\).
  • Final layer: Multiply the latest result (27) by the base one last time: \(27 \times 3 = 81\).
Each step makes use of the previous result, making sure we multiply exactly four times, following the exponent's instruction.
Simplification Steps
The simplification of \(3^4\) into 81 involves careful steps of breaking down the multiplication process. By understanding these steps, the problem transforms from a complex expression into simple arithmetic.
  • Have clarity on the base and exponent.
  • Remember to multiply in layers, one result building upon the last.
  • Carefully follow through each multiplication to ensure accuracy in simplification.
Breaking down \(3^4\) to its simplest form of 81 can help demystify exponentiation. Each step in the process is deliberate and helps ensure you understand not just the 'what' but the 'why' behind each calculation. Remember: practice makes everything easier to grasp.

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Most popular questions from this chapter

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{225}{15}$$

Write each mixed number as an improper fraction. See Example 20. $$166 \frac{2}{3}$$

Recall that to find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of \(\frac{1}{2}\) and \(\frac{3}{4}\) is \(\frac{\frac{1}{2}+\frac{3}{4}}{2}\). Find the average of each pair of numbers. $$\frac{1}{4}, \frac{2}{14}$$

Use the following numbers: \(\begin{array}{llllllll}8691 & 786 & 1235 & 2235 & 85 & 105 & 22 & 222 & 900 & 1470\end{array}\); List the numbers divisible by both 2 and \(3 .\)

Write each fraction in two other equivalent ways by inserting the negative sign in different places. $$-\frac{45}{-57}=\quad=$$
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