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Chapter 5: Problem 50

Express each number in decimal notation. \(1 \times 10^{8}\)

Short Answer

Expert verified
The decimal notation for \(1 \times 10^{8}\) is 100000000.

Step by step solution

01

Understanding the Power of 10

Observe the number \(1 \times 10^{8}\). The exponent of 10 is 8, which is a positive number. It means we should move the decimal point 8 places to the right.
02

Applying the Power of 10

Starting from the number 1 and moving the decimal point 8 places to the right, we obtain 100000000. If there are not enough numbers, zeroes are added to compensate.

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Key Concepts

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

Powers of Ten
The concept of powers of ten refers to using the number ten raised to an exponent. This is a nifty way to represent numbers that are either very large or very small in a compact form. A power of ten is written as \(10^n\), where \(n\) is the exponent. When the exponent, \(n\), is positive, it tells us how many zeros follow the number one. For example, \(10^3\) means "1 followed by 3 zeros," which is 1000.
Powers of ten make it easy to quickly calculate or understand the scale of a number. Knowing how to use powers of ten effectively helps in converting between standard and scientific notation.
Exponents
Exponents tell us how many times a number, known as the base, is multiplied by itself. In the expression \(10^8\), 10 is the base and 8 is the exponent. So, \(10^8\) means multiplying 10 by itself 8 times:
  • \(10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10\)
  • This results in 100,000,000.
The usage of exponents is vital in mathematics as it simplifies the representation of large numbers and performs extensive calculations quickly.
Scientific Notation
Scientific notation is a method used to write very large or very small numbers succinctly. It is especially useful in science and engineering because these fields often deal with extreme values. In scientific notation, a number is expressed as the product of a number between 1 and 10, and a power of ten. For instance, the number 100,000,000 can be written in scientific notation as \(1 \times 10^8\). This form is useful as it makes numbers more manageable for calculations and comparisons.
Convert to Decimal
Converting a number from scientific notation to decimal notation involves moving the decimal point. When converting \(1 \times 10^8\) to decimal, identify the exponent (which is 8) and move the decimal point in the number 1 a total of 8 places to the right. As you move the decimal point, fill in with zeroes if necessary:
  • Starting from 1, you move the decimal 8 places to the right.
  • This gives you 100,000,000 in decimal form.
By mastering this process, you'll find it easier to switch between scientific and decimal notations as needed.

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Most popular questions from this chapter

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(3,8,13,18, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{40}\), when \(a_{1}=1000, r=-\frac{1}{2}\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.

In Exercises 49-70, write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=4, r=2\)

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(3,15,75,375, \ldots\)
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