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Chapter 6: Problem 572

Divide. Write your answer in decimal form. $$\frac{6 \times 10^{4}}{3 \times 10^{-2}}$$

Short Answer

Expert verified
2000000

Step by step solution

01

- Understand the Expression

The expression to divide is \(\frac{6 \times 10^{4}}{3 \times 10^{-2}}\). This involves both a numerical coefficient and a power of 10.
02

- Divide the Numerical Coefficients

Divide the coefficients: \( \frac{6}{3} = 2 \).
03

- Apply the Laws of Exponents

When dividing exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator: \((10^{4})/(10^{-2}) = 10^{4-(-2)} = 10^6\).
04

- Combine Coefficients and Powers of 10

Combine the results from the numerical division and the exponent rules: \(2 \times 10^6\).
05

- Convert to Decimal Form

Convert \(2 \times 10^6\) to its decimal form: 2000000.

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Key Concepts

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

Division
Division is the process of determining how many times one number, called the divisor, is contained within another number, called the dividend. In this exercise, we handle division with both numerical coefficients and exponents. The first step in our example is to divide the numerical coefficients: \( \frac{6}{3} = 2 \). This simplifies the process, allowing us to focus on the exponents next. Breaking the problem into smaller parts makes it easier to handle.
Laws of Exponents
The laws of exponents are critical when dividing numbers in scientific notation. The most important rule here is when dividing two powers of 10: \( \frac{10^a}{10^b} = 10^{a-b} \). In our sample problem, we apply this by subtracting the exponents \( 4 - (-2) = 6 \). This turns our expression into \( 10^6 \). Remember, subtracting a negative exponent is the same as adding the absolute value of that exponent.
Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a compact form. A number in scientific notation is written as the product of a coefficient and a power of 10. In our problem, after applying the laws of exponents we get \( 2 \times 10^6 \). This format is very helpful when dealing with astronomical or microscopic values, making calculations simpler and reducing errors.
Decimal Conversion
Once we have our answer in scientific notation, we may need to convert it to decimal form to make it easier to read or understand. To do this, we shift the decimal point of the coefficient based on the exponent. For example, \( 2 \times 10^6 \) means moving the decimal point in '2' six places to the right, resulting in 2,000,000. This step is crucial for interpreting the result in a more familiar decimal notation.

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Most popular questions from this chapter

Mixed Practice (a) \(q^{5} \cdot q^{3}\) (b) \(\left(q^{5}\right)^{3}\)

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Divide \(\frac{10 x^{2}+x-12}{2 x}\) and explain with words how you get each term of the quotient.

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