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Chapter 0: Problem 6

In Exercises \(1-28\), compute the numbers. $$ (100)^{4} $$

Short Answer

Expert verified
The result of \( (100)^4 \) is 100,000,000.

Step by step solution

01

Understanding the Problem

The problem requires calculating \( (100)^{4} \). This means raising 100 to the power of 4.
02

Applying the Exponent Rule

To solve this, we use the exponentiation rule, which states that \( a^b \) means multiplying a by itself b times.
03

Perform the Calculation

Calculate \( 100^4 \) by multiplying 100, four times: \[ 100 \times 100 \times 100 \times 100 \].
04

Simplify the Multiplication

Perform the multiplication step-by-step: \[ 100 \times 100 = 10,000 \] \[ 10,000 \times 100 = 1,000,000 \] \[ 1,000,000 \times 100 = 100,000,000 \]. Therefore, \( 100^4 = 100,000,000 \).

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Key Concepts

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

power rule
Exponentiation involves raising a number to a power. The power rule, given by \( a^b \), means multiplying the base \(a\) by itself \(b\) times. This rule is foundational in mathematics. For instance, in the exercise, we calculated \( 100^4 \), which means \(100\) multiplied by itself four times: \[ 100 \times 100 \times 100 \times 100 \].The power rule helps simplify complex multiplication, especially with large numbers.

By understanding this rule, you can tackle a wide range of exponentiation problems, making calculations more manageable and systematic.
multiplication
Multiplication is a basic arithmetic operation that combines equal groups. In exponentiation, it simplifies repeated addition. When raising a number to a power, multiplication becomes essential.

For example, to compute \( 100^4 \), we multiply \(100\) by itself multiple times:
  • First, calculate \( 100 \times 100 = 10,000 \)
  • Then, \( 10,000 \times 100 = 1,000,000 \)
  • Finally, \( 1,000,000 \times 100 = 100,000,000 \)
Breaking down the steps highlights the importance of multiplication in solving exponentiation problems efficiently.

This method not only ensures accuracy but also provides a clear approach to handling large numbers.
large numbers
Working with large numbers can be intimidating, but understanding exponentiation and multiplication makes it manageable. In our example, computing \( 100^4 \) yields \( 100,000,000 \), which is a large number. To handle such calculations:
  • Break down the problem into smaller steps.
  • Use the power rule to simplify the process.
  • Apply multiplication methodically to ascertain accuracy.
Dealing with large numbers requires patience and practice. By breaking the calculations into manageable parts, you can avoid errors and understand the underlying math concepts better.

Engaging with large numbers is a valuable skill, especially in areas such as science, engineering, and finance.

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Most popular questions from this chapter

Let \(f(x)=x^{2}\). Graph the functions \(f(x+1), f(x-1)\), \(f(x+2)\), and \(f(x-2)\). Make a guess about the relationship between the graph of a general function \(f(x)\) and the graph of \(f(g(x))\), where \(g(x)=x+a\) for some constant \(a\). Test your guess on the functions \(f(x)=x^{3}\) and \(f(x)=\sqrt{x}\).

If \(f(x)=1 / x\), find \(f(x+h)-f(x)\) and simplify.

Use the laws of exponents to compute the numbers. \(\left(3^{1 / 3} \cdot 3^{1 / 6}\right)^{6}\)

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}}\). Calculate the following functions. Take \(x>0\). \(\frac{g(x)}{f(x)}\)

In Exercises find a good window setting for the graph of the function. The graph should show all the zeros of the polynomial. $$ f(x)=x^{4}-200 x^{3}-100 x^{2} $$
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