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Chapter 5: Problem 168

Convert each number to decimal form. (a) \(7.5 \times 10^{6}\) (b) \(-4.13 \times 10^{-5}\)

Short Answer

Expert verified
a) 7,500,000 b) -0.0000413

Step by step solution

01

Understand Scientific Notation

Scientific notation is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal form. Numbers are written as a product of a coefficient and a power of 10.
02

Convert (a) 7.5 x 10^6

To convert the number to decimal form, move the decimal point 6 places to the right, as the exponent is positive: 7.5 becomes 7500000. Thus, 7.5 x 10^6 in decimal form is 7,500,000.
03

Convert (b) -4.13 x 10^-5

To convert the number to decimal form, move the decimal point 5 places to the left, as the exponent is negative: -4.13 becomes -0.0000413. Thus, -4.13 x 10^-5 in decimal form is -0.0000413.

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Key Concepts

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

positive exponents
To understand positive exponents, we need to grasp what an exponent signifies. The exponent tells us how many times to multiply the base number by 10. When you have a number like \(7.5 \times 10^6\), the exponent is 6. This means you move the decimal point 6 places to the right.
Starting with 7.5, you move the decimal point:
  • First move: 75.0
  • Second move: 750.0
  • Third move: 7500.0
  • Fourth move: 75000.0
  • Fifth move: 750000.0
  • Sixth move: 7500000.0

Thus, \(7.5 \times 10^6\) becomes 7,500,000 in decimal form. Remember, a positive exponent shifts the decimal to the right.
negative exponents
Negative exponents work oppositely compared to positive ones. A negative exponent indicates that we need to divide by 10, the number of times specified by the exponent. This essentially moves the decimal point to the left.
For example, with \(-4.13 \times 10^{-5}\), the exponent is -5, meaning the decimal point moves 5 places to the left.
  • First move: -0.413
  • Second move: -0.0413
  • Third move: -0.00413
  • Fourth move: -0.000413
  • Fifth move: -0.0000413
So, \(-4.13 \times 10^{-5}\) turns into -0.0000413 in decimal form. Essentially, negative exponents make numbers smaller by shifting the decimal point leftward.
decimal conversion
Converting scientific notation to decimal form involves shifting the decimal based on the exponent. Here鈥檚 how you can do it:
To move the decimal point:
  • Count the number of spaces needed according to the exponent.
  • Move the decimal point accordingly.

For positive exponents, move the decimal to the right as many times as the exponent. For negative exponents, shift it to the left that many times.
Consider the example \(7.5 \times 10^6\). Since the exponent is 6, we move the decimal 6 places to the right to get 7,500,000.
For \(-4.13 \times 10^{-5}\), move the decimal 5 places to the left, resulting in -0.0000413. By diligently following these steps, you can convert any number from scientific notation to its decimal form.

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