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Chapter 14: Problem 53

Convert to scientific notation. $$0.000000437$$

Short Answer

Expert verified
The short answer is: \(4.37 脳 10^{-7}\).

Step by step solution

01

Identify a and n

We want to rewrite the number 0.000000437 as $$a 脳 10^n$$ First, we need to create a number between 1 and 10 by moving the decimal point. For 0.000000437, if we move the decimal point 7 places to the right, we get \(a = 4.37\). Now we need to find the appropriate power of 10. Since we moved the decimal point 7 places to the right, \(n = -7\). Thus, the desired scientific notation is: $$4.37 脳 10^{-7}$$.

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Key Concepts

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

Decimal Point
The decimal point is a crucial part of understanding decimal numbers and scientific notation. In decimal notation, it separates the integer part of a number from the fractional part. For example, in the number 0.000000437, the decimal point separates the leading zeros from the non-zero digits.

When converting to scientific notation, adjusting the decimal point is vital for creating a manageable number. We want a number, known as 'a', that lies between 1 and 10. To achieve this, you adjust the position of the decimal point.

In 0.000000437, moving the decimal point 7 places to the right yields 4.37, as required for scientific notation.
Powers of 10
Powers of 10 are essential when converting numbers into scientific notation. These powers allow us to express very large or very small numbers succinctly using exponents. In this context, a power of 10 indicates how many times you multiply the base number, 10, by itself.

Positive exponents represent large numbers, while negative exponents denote small numbers. For 0.000000437, moving the decimal point to form the number 4.37 required shifting it 7 places to the right.

This movement is represented by the power \(10^{-7}\), indicating that 4.37 is multiplied by \(10^{-7}\), to accurately express the original number.
Conversion Process
The conversion process to scientific notation involves a few straightforward steps. First, identify a number between 1 and 10, which will be the result of adjusting the decimal point position. In the example 0.000000437, moving the decimal point 7 places to the right gives you 4.37.

Next, determine the power of 10. This power is equal to the number of places you moved the decimal point. Moving it to the right means you'll have a negative exponent. Here, moving 7 places gives \(n = -7\).

Finally, combine these elements into the scientific notation format, \(a \times 10^n\). So, 0.000000437 becomes \(4.37 \times 10^{-7}\), compactly representing your original number.

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Most popular questions from this chapter

The three foreign countries from which the largest number of children were adopted in 2009 were China, Ethiopia, and Russia. A total of 6864 children were adopted from these countries. The number of children adopted from China was 862 fewer than the total number adopted from Ethiopia and Russia. Twice the number adopted from Russia is 171 more than the number adopted from China. (Source: U.S. Department of State) Find the number of children adopted from each country.

Approximate the point \((s)\) of intersection of the pair of equations. $$y=2.3 \ln (x+10.7), y=10 e^{-0.07 x^{2}}$$

Solve the system of equations. $$\begin{aligned} 2 x+y-3 z &=1 \\ x-4 y+z &=6 \\ 4 x-7 y-z &=13 \end{aligned}$$

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log _{5} x=4$$

The top three Mother's Day gifts are flowers, jewelry, and gift certificates. The total of the average amounts spent on these gifts is \(\$ 53.42 .\) The average amount spent on jewelry is \(\$ 4.40\) more than the average amount spent on gift certificates. Together, the average amounts spent on flowers and gift certificates is \(\$ 15.58\) more than the average amount spent on jewelry. (Source: BIG research) What is the average amount spent on each type of gift?
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