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Chapter 3: Problem 34

Simplify. \(\frac{1.2 \times 10^{3}}{2.4 \times 10^{-17}}\)

Short Answer

Expert verified
5 \times 10^{19}

Step by step solution

01

Understand the Problem

The goal is to simplify the expression \( \frac{1.2 \times 10^{3}}{2.4 \times 10^{-17}} \). This is a problem involving division of numbers in scientific notation.
02

Divide the Coefficients

Divide the coefficients (the numbers before the powers of ten): \( \frac{1.2}{2.4} = 0.5 \).
03

Subtract the Exponents

Using the property of exponents, when dividing like bases, subtract the exponents: \( 10^{3} / 10^{-17} = 10^{3 - (-17)} = 10^{3 + 17} = 10^{20} \).
04

Combine the Results

Multiply the simplified coefficient by the simplified power of ten: \( 0.5 \times 10^{20} \).
05

Express in Proper Scientific Notation

In proper scientific notation, move the decimal place in the coefficient if necessary: \( 0.5 \times 10^{20} = 5 \times 10^{-1} \times 10^{20} = 5 \times 10^{19} \).

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Key Concepts

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

simplifying scientific notation
Simplifying scientific notation involves breaking down a complex expression into a more manageable form. Here, we divide the coefficients and then subtract the exponents of the powers of ten.

When dealing with scientific notation:
  • First, handle the coefficients. For example, from the exercise, we divided the coefficients to get: \( \frac{1.2}{2.4} = 0.5 \).
  • Next, we work with the exponents by subtracting them, as with: \( 10^{3} - (-17) = 10^{20} \).
Finally, you combine these results to simplify the expression into proper scientific notation.
properties of exponents
The properties of exponents are crucial when working with scientific notation. One of the core principles is that when you divide powers of the same base, you subtract the exponents.

This is reflected in the exercise:
  • Given: \( 10^{3} / 10^{-17} \).
  • Using the property, we subtract the exponents: \( 3 - (-17) \).
  • This simplifies to: \( 10^{20} \).
Understanding these basic properties will make complex scientific notation problems much simpler to solve.
proper scientific notation
To express a number in proper scientific notation, your coefficient must be between 1 and 10. This means you sometimes need to adjust the coefficient and exponent accordingly.

In the exercise, our result was initially \( 0.5 \times 10^{20} \). Since 0.5 is not between 1 and 10, we move the decimal point to get a proper coefficient of 5.

By doing this, you must also adjust the exponent, making:
  • \( 0.5 \times 10^{20} \) into \( 5 \times 10^{-1} \times 10^{20} \).
  • Simplify it to: \( 5 \times 10^{19} \).
Proper scientific notation ensures consistency and clarity in mathematical expressions.

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Most popular questions from this chapter

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