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Chapter 5: Problem 74

\(3.9 \times 10^{-6} \mathrm{~m}+4.2 \times 10^{-6} \mathrm{~m}\)

Short Answer

Expert verified
The result is \(8.1 \times 10^{-6} \mathrm{~m}\).

Step by step solution

01

- Understand the Problem

We need to add two numbers: \(3.9 \times 10^{-6} \mathrm{~m}\) and \(4.2 \times 10^{-6} \mathrm{~m}\). These are numbers in scientific notation with the same exponent.
02

- Add the Numbers

Since the exponents are the same, we directly add the coefficients (the numbers in front of the exponential part). So, we add 3.9 and 4.2.
03

- Perform the Addition

\(3.9 + 4.2 = 8.1\).
04

- Write the Result in Scientific Notation

After adding the coefficients, we keep the exponent the same: \(8.1 \times 10^{-6} \mathrm{~m}\).

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Key Concepts

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

scientific notation
Scientific notation is a way to express really big or super tiny numbers using powers of ten. It's very common in science and math. For instance, instead of writing 0.0000012, you can write it as \(1.2 \times 10^{-6}\). This simply means that you move the decimal point six places to the left.
There's a basic structure to scientific notation:
  • A coefficient that's at least 1 but less than 10
  • Followed by a multiplication sign
  • And then a power of ten

When you see something like \(3.9 \times 10^{-6} \text{m}\), think of it as 3.9 multiplied by a tiny number, which is why it's negative because it tells you to move the decimal point to the left.
addition of exponents
Adding numbers in scientific notation can be straightforward when their exponents (the power of 10 part) are the same. For example, \(3.9 \times 10^{-6}\) and \(4.2 \times 10^{-6}\) both have \(10^{-6}\). When the exponents match, it's a breeze:
  • Just add the coefficients together
  • Keep the exponent as it is

This means for our example: \(3.9 + 4.2 = 8.1\). The exponents stay the same, so the final answer looks like \(8.1 \times 10^{-6} \text{m}\). Remember, you don鈥檛 change the exponent during the addition.
coefficients
In scientific notation, the coefficient is the number in front of the power of ten. It's what you see first and it's always between 1 and 10. For example, in \(3.9 \times 10^{-6}\), the coefficient is 3.9, and in \(4.2 \times 10^{-6}\), the coefficient is 4.2.
When you're adding numbers in scientific notation with the same exponent, you sum up these coefficients:
  • First, identify the coefficients (3.9 and 4.2)
  • Add them together (3.9 + 4.2)
  • Write down the result (8.1)

Pair this sum with the shared exponent to get your final result: \(8.1 \times 10^{-6} \text{m}\). This makes adding in scientific notation quick and efficient.

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Most popular questions from this chapter

The base of a triangle is 4 in. longer than twice the height. a. If \(h=\) height, write a polynomial expression in \(h\) that represents the base, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(h\) that represents the area.

\(\left(y^{2}-6 y+9\right) \div(3 y)\)

\(\left(x^{2}+13 x+18\right) \div(x-3)\)

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\(\left(2 x^{2}-x-15\right) \div(2 x+5)\)
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