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Chapter 9: Problem 124

For Exercises, write in decimal notation. $$2 \times 10^{15}$$

Short Answer

Expert verified
The decimal notation of \(2 \times 10^{15}\) is 2,000,000,000,000,000.

Step by step solution

01

Identify the Base and Exponent

In the given expression \(2 \times 10^{15}\), the base number is 2 and the exponent is 15.
02

Move the Decimal Point

Since the exponent is positive, move the decimal point in the base number 15 places to the right. If the base number doesn't have enough digits, add zeros.
03

Write in Decimal Notation

After moving the decimal point 15 places to the right, the number in decimal notation is 2,000,000,000,000,000.

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Key Concepts

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

Exponents
Exponents are a mathematical shorthand used to express how many times a number, known as the base, is multiplied by itself. For example, in the expression \(2^{15}\), 2 is the base and 15 is the exponent.
This means you multiply the base, 2, by itself, 15 times. It is a powerful and efficient way to represent very large numbers without writing all the multiplication steps.
  • In the expression \(2 \times 10^{15}\), the exponent 15 applies to the base 10. It indicates that you multiply 10 by itself 15 times.
  • This results in 10,000,000,000,000,000, effectively a 1 followed by 15 zeros.
Understanding exponents is crucial because they form the foundation for more advanced topics like scientific notation and logarithms.
Scientific Notation
Scientific notation is a convenient way of expressing very large or very small numbers in a concise format. It's commonly used in scientific and engineering fields to simplify calculations and reduce errors.
In scientific notation, a number is written as the product of two parts:
  • A number greater than or equal to 1 but less than 10.
  • And an exponent of 10.
For instance, \(2 \times 10^{15}\) is already in scientific notation. Here, 2 is the coefficient, and \(10^{15}\) shows how many places the decimal point has moved.
This representation efficiently compresses a large number into a shorter form, making it more manageable for computations. To convert it back to decimal form, you'd move the decimal 15 places to the right.
Place Value
Place value is the foundation of our number system and helps us understand the value of digits in a number based on their position.
The place of a digit in a number determines how large or small the value of that digit is.
For example, in 2,000,000,000,000,000, each zero changes the scale of the number by a factor of 10, based on its position relative to the base.
  • In general, starting from the right, the first position is the "ones" place; the next is the "tens" place, then "hundreds," "thousands," and so on.
  • The pattern continues, with each new place being ten times the place to its right.
In \(2 \times 10^{15}\), we add 15 zeros after the 2, which pushes the digit 2 into the 10 quadrillion place, illustrating the power of place value in representing very large numbers.

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