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Chapter 5: Problem 29

\(\frac{6 \times 10^{-8}}{2 \times 10^{3}}\)

Short Answer

Expert verified
\(3 \times 10^{-11}\)

Step by step solution

01

Understand the Problem

The given expression is \(\frac{6 \times 10^{-8}}{2 \times 10^{3}}\). This is a division problem involving scientific notation.
02

Divide the Coefficients

To divide numbers in scientific notation, first divide the coefficients (the numbers in front). So, divide 6 by 2. \[\frac{6}{2} = 3\]
03

Subtract the Exponents

Next, subtract the exponents. The exponents are -8 and 3, so \(-8 - 3 = -11\).
04

Combine Results

Combine the result of the coefficients and the exponents. The final answer is \(3 \times 10^{-11}\).

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Key Concepts

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

Dividing Scientific Notation
Scientific notation is a concise way to express very large or small numbers. It consists of two parts: a coefficient and an exponent with base 10. Dividing numbers in scientific notation simplifies when you break the problem into smaller steps.

To divide numbers in scientific notation, you follow these steps:
  • First, divide the coefficients (the numbers in front of the base 10).
  • Then, subtract the exponents.
Let's consider the example: \(\frac{6 \times 10^{-8}}{2 \times 10^{3}}\)

First, divide the coefficients 6 by 2, resulting in 3. Next, subtract the exponents: (-8) - 3, which gives -11. Lastly, combine these results to get the final answer: \[3 \times 10^{-11}\]
Exponents
Understanding exponents is essential when working with scientific notation. An exponent shows how many times the base (usually 10) is multiplied by itself. For example, \(10^3 = 10 \times 10 \times 10 = 1000\)

When dividing scientific notations, you subtract the exponent of the denominator from the exponent of the numerator. In our example, we have \(10^{-8}\) and \(10^3\). Subtract the exponent of the denominator from the exponent of the numerator: \(-8 - 3 = -11\).

Therefore, understanding how to operate with exponents helps simplify and solve problems more efficiently. This becomes especially handy when dealing with extremely large or small numbers.
Coefficients
Coefficients are the numbers in front of the base 10 in scientific notation. They are dynamic and can be easily multiplied, divided, added, or subtracted, making them a crucial part of simplifying expressions.

In our example, we had to divide the coefficients. Given two coefficients, 6 and 2, the division really becomes straightforward: \( \frac{6}{2} = 3 \). This step essential because it simplifies the overall calculation.

Coefficients make scientific notation manageable, compressing vast quantities of information into simple, usable terms. Remember, no matter how tiny or massive a number is, coefficients play a pivotal role in working with scientific notation formats.

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Problem: Simplify: \((3 p)(p+4)(p+2)\) $$ \text { Incorrect Answer: } \begin{aligned} &(3 p)(p+4)(p+2) \\ &=3 p(p+4)+3 p(p+2) \\ &=3 p(p)+3 p(4)+3 p(p)+3 p(2) \\ &=3 p^{2}+12 p+3 p^{2}+6 p \\ &=6 p^{2}+18 p \end{aligned} $$

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