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Chapter 1: Problem 18

Find the value of each exponential expression. $$ 6^{4} $$

Short Answer

Expert verified
1296

Step by step solution

01

Understand the Exponential Expression

The expression given is an exponential expression: \(6^{4}\). It means that the base number 6 is multiplied by itself 4 times.
02

Write the Expression in Expanded Form

Expand the expression \(6^{4}\) as follows: \(6 \times 6 \times 6 \times 6\).
03

Perform the Multiplication

Calculate the value step by step: \(6 \times 6 = 36\) \(36 \times 6 = 216\) \(216 \times 6 = 1296\).
04

State the Final Answer

After performing the multiplications, the value of \(6^{4}\) is 1296.

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number that is being multiplied, while the exponent tells us how many times to multiply the base by itself. For example, in the expression \(6^{4}\), 6 is the base, and 4 is the exponent.

This means we must multiply 6 by itself 4 times. In more general terms, \(a^{n}\) means that a is multiplied by itself n times, which provides the basis for many more complex mathematical computations.
  • The base number: The number being multiplied.
  • The exponent: Indicates how many times to multiply the base number.
  • Result: The final product after repeated multiplication.
Expanded Form
Breaking down an exponential expression into its expanded form helps in visualizing and understanding the multiplication process. Going back to our example \(6^{4}\), we can write this in expanded form as:
\(6 \times 6 \times 6 \times 6\)

When expressed in this way, it is easier to see that we are multiplying the base number (6) four times.

  • Helps to break down the problem into smaller, more manageable steps.
  • Makes the multiplication process clear and straightforward.
  • Allows you to perform each multiplication individually, reducing the risk of mistakes.
Multiplication
Multiplication, especially in the context of exponentiation, is about repeated addition. With the expanded form, we can multiply step-by-step for better understanding. Taking our example further:

Step 1: \(6 \times 6 = 36\)

Step 2: \(36 \times 6 = 216\)

Step 3: \(216 \times 6 = 1296\)
  • Makes large calculations simpler by breaking them into small steps.
  • Prevents errors in computation by ensuring each step is done correctly.
  • Strengthens understanding of basic operations and their properties.
Finally, multiplying in steps can make daunting problems much more approachable.
After all the steps, you find the solution of \(6^{4}\) to be 1296.

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Most popular questions from this chapter

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 9 if the sum of its digits is divisible by \(9,\) and not otherwise. Show that (a) \(4,114,107\) is divisible by \(9 \quad\) and \(\quad\) (b) \(2,287,321\) is not divisible by \(9 .\)

Perform each indicated operation. $$ 6-(-8+3) $$

Solve each problem. Bruce Buit also enjoys playing Triominoes. On five successive turns, his scores were \(-13,15,-12,24,\) and \(14 .\) What was his total score for the five turns?

Perform each indicated operation. $$ -4+[(-12+1)-(-1-9)] $$

Decide whether each statement is true or false. \(-|-12| \leq-|-15|\)
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