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Chapter 2: Problem 30

Write each number in scientific notation. 62,000

Short Answer

Expert verified
The scientific notation of 62,000 is \(6.2 \times 10^4\).

Step by step solution

01

Determine the Decimal Number Between 1 and 10

The given number is 62,000. To find the decimal number between 1 and 10, move the decimal point to the left until there is only one non-zero digit to the left of the decimal point. In our case, move the decimal point 4 places to the left: \[62,000 = 6.2 \times 10^4\] So, the decimal number between 1 and 10 is 6.2.
02

Calculate the Power of 10

Now we need to determine the power of 10 that we need to multiply 6.2 with to get the original number. We moved the decimal point 4 places to the left, so the power of 10 is 4. So, the scientific notation of 62,000 is: \[6.2 \times 10^4\]

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Key Concepts

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

Decimal Conversion
When dealing with large numbers like 62,000, converting them to a simpler form makes it easier to handle and understand. One step in this process is decimal conversion. This means adjusting the number into a form that has a decimal point placed after the first non-zero digit.

To do this for 62,000, we imagine where the decimal would be (right after the last zero). By moving the decimal four spaces to the left, we change 62,000 into 6.2. This process drastically simplifies the number by creating a smaller, more manageable figure between 1 and 10.

Remember:
  • Start with the entire number and locate where the decimal would logically be.
  • Shift the decimal left until you find yourself with one non-zero digit in front.
  • The final product should be a number between 1 and 10.
Decimal conversion lays the foundation for the number to be expressed in scientific notation.
Power of 10
Next, let's consider the power of 10. When converting a large number to scientific notation, it's all about understanding how to "scale" the already converted decimal back to its original size. This is where the power of 10 plays its role.

After converting 62,000 into 6.2 through decimal conversion, we need to determine how many places the decimal was moved. In this case, we moved it four times to the left. This movement is represented as a power of 10, specifically, \[10^4\].

Key points:
  • The number of decimal places you move becomes your exponent in the power of 10.
  • Left movement of the decimal results in a positive exponent, while a right movement would instead give a negative exponent.
  • This exponent dictates how large or small the number is in relation to its decimal conversion.
Understanding powers of 10 allows us to multiply the simplified decimal, 6.2, back to its original scale, hence forming the scientific notation.
Writing Numbers in Standard Form
Once you grasp both decimal conversion and powers of 10, you can put these elements together to write numbers in scientific notation, also known as standard form.

Scientific notation provides a streamlined way to represent large numbers, such as 62,000. By expressing it as \[6.2 \times 10^4\],we create a brief yet powerful way to convey the same information.

Steps involved:
  • Convert the number to a decimal between 1 and 10.
  • Determine the necessary power of 10 based on decimal places moved.
  • Combine these into a single multiplication expression.
This form articulates the essential magnitude and scale of the original number efficiently. Scientific notation eliminates unnecessary zeros and reduces the space needed to represent huge or tiny figures, all while maintaining accuracy and precision.

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Most popular questions from this chapter

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{6 \times 10^{9}}{2 \times 10^{5}}\)

Simplify the expression using the product rule. Leave your answer in exponential form. $$k \cdot k^{2} \cdot k^{3}$$

Simplify the expression using one of the power rules. $$\left(-5^{3}\right)^{2}$$

Simplify the expression using the product rule. Leave your answer in exponential form. $$\left(\frac{8}{21} b\right)\left(-6 b^{8}\right)\left(-\frac{7}{2} b^{6}\right)$$

Simplify using the quotient rule. $$\frac{63 a^{-2} b^{2}}{9 a^{7} b^{10}}$$
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