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Chapter 5: Problem 104

Write each number in scientific notation. See Example 8. $$ 36,800,000 $$

Short Answer

Expert verified
36,800,000 is written as \( 3.68 \times 10^7 \) in scientific notation.

Step by step solution

01

Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small in a compact form that is usually written as \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.
02

Identify the Significant Digits

For the number 36,800,000, identify the significant digits that are not zero. This is 3.68, because trailing zeroes in a large number are not significant by themselves in scientific notation.
03

Determine the Power of 10

Count how many places the decimal point needs to be moved to place it after the first digit. For 36,800,000, the decimal point moves 7 places to the left to be between 3 and 6.
04

Write in Scientific Notation

Combine the significant digits and power of 10 to rewrite the number. Thus, 36,800,000 can be expressed as \( 3.68 \times 10^7 \).

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Key Concepts

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

Significant Digits
Understanding significant digits is crucial in scientific notation. They are the digits in a number that carry meaningful contributions to its measurement accuracy. Simply put, they tell us which numbers are essential in expressing the true value of the given number. Consider the number 36,800,000; here, 3, 6, and 8 are the significant digits.
  • Zeroes between non-zero digits are always significant.
  • Leading zeroes in numbers less than one are not significant.
  • In large numbers, trailing zeroes are not significant unless specified by a decimal point.
In the number 36,800,000, even though there are many zeroes, the significant digits are 3.68. This is because, in scientific notation, these digits describe the number precisely. Always pay attention to these rules when identifying significant digits in a number.
Power of 10
The power of 10 in scientific notation indicates how many places the decimal point has been moved. It allows us to handle both very large and very small numbers comfortably.This notation consists of two parts:
  • The coefficient or base: A number between 1 and 10
  • The exponent of 10: Represents the number of places the decimal moves
For the number 36,800,000, the decimal must move 7 places to the left to position it between the first two digits, 3 and 6. This gives us a power of 10 as 7, making the scientific notation look like this: \[3.68 \times 10^7\] Here, the exponent 7 signifies that we have shifted the decimal 7 places left to express the number optimally.
Large Numbers
Large numbers can be complex and cumbersome to read, write, and calculate. Scientific notation helps simplify these tasks and makes managing large values much more accessible. Numbers like 36,800,000 become less intimidating.Imagine writing or even reading such long numbers repeatedly. Instead of all those zeroes, scientific notation lets us summarize this value as \(3.68 \times 10^7\). It鈥檚 both compact and informative.
  • Easier to perform arithmetic operations like multiplication and division
  • Reduces error risk in reading and writing the number
Scientific notation excels in fields like science and engineering, where large numbers are common, and accuracy is paramount. By understanding and employing scientific notation, you can better tackle challenges involving both large and small numbers.

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Most popular questions from this chapter

Which numbers are equal to \(36,000 ?\) Of these, which is written in scientific notation? $$a. 36 \times 10^{3}$$ $$b.360 \times 10^{2}$$ $$c. 0.36 \times 10^{5}$$ $$d. 3.6 \times 10^{4}$$

Factor each polynomial completely. See Examples 1 through 12. $$ 18 x^{4}+21 x^{3}+6 x^{2} $$

If \(P(x)\) is the polynomial given, find a. \(P(a),\) b. \(P(-x),\) and c. \(P(x+h)\). \(P(x)=3 x-2\)

Factor. Assume that variables used as exponents represent positive integers. $$ 9 x^{2 n}+24 x^{n}+16 $$

Factor each polynomial completely. See Examples 1 through 12. $$ 2 x^{3} y+2 x^{2} y-12 x y $$
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