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Chapter 7: Problem 52

Do your computation using scientific notation. $$\frac{80}{0.0002}$$

Short Answer

Expert verified
4.0 x 10^5

Step by step solution

01

Express the Numbers in Scientific Notation

To convert 80 into scientific notation, write it as \( 8.0 \times 10^1 \). For 0.0002, write it as \( 2.0 \times 10^{-4} \). Now the problem becomes \( \frac{8.0 \times 10^1}{2.0 \times 10^{-4}} \).
02

Simplify the Fraction

Write the fraction as two separate parts: the coefficient and the power of ten. So, \( \frac{8.0}{2.0} \times \frac{10^1}{10^{-4}} \).
03

Divide the Coefficients

Divide \( 8.0 \) by \( 2.0 \), which equals 4. So now we have \( 4.0 \times \frac{10^1}{10^{-4}} \).
04

Divide the Powers of Ten

Apply the rule for dividing exponents: \( \frac{10^1}{10^{-4}} = 10^{1 - (-4)} = 10^{1 + 4} = 10^5 \). Now the expression is \( 4.0 \times 10^5 \).

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Key Concepts

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

Elementary Algebra
Elementary algebra is the foundation for understanding and manipulating mathematical expressions and equations. It involves using variables to represent numbers and creating relationships between them. In our exercise, we started by converting both numbers - 80 and 0.0002 - into scientific notation. This is a common technique in algebra to simplify complex numbers and make calculations more manageable. Converting the numbers to scientific notation makes the problem straightforward: we work with smaller, easily understandable units.
Powers of Ten
Understanding powers of ten is crucial when working with scientific notation. Powers of ten describe how many times a number is multiplied by itself. For instance, multiplying by 10 repeatedly (like 10, 100, 1000) means using powers of ten. In our example, we transformed 80 to \( 8.0 \times 10^1 \) and 0.0002 to \( 2.0 \times 10^{-4} \). These transformations make it easier to handle large or tiny numbers. When dividing powers of ten, we subtract the exponents: \[ \frac{10^1}{10^{-4}} = 10^{1 - (-4)} = 10^{1 + 4} = 10^5 \]. This knowledge helps us simplify scientific notation problems without confusion.
Simplification of Fractions
Simplification of fractions is about making fractions easier to work with by reducing them to their simplest form. In our problem, after converting to scientific notation, we had \( \frac{8.0 \times 10^1}{2.0 \times 10^{-4}} \). We simplified by dealing with the coefficients first: \( \frac{8.0}{2.0} = 4 \). Next, we handled the powers of ten: \( \frac{10^1}{10^{-4}} = 10^5 \). So, the final simplified expression is \( 4.0 \times 10^5 \). Always break down more complex fractions into smaller, simpler parts for easier solving.

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