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Chapter 10: Problem 45

For Exercises \(33-48,\) convert to scientific notation. $$0.082$$

Short Answer

Expert verified
The scientific notation of 0.082 is \( 8.2 \times 10^{-2} \).

Step by step solution

01

Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is generally expressed as \[ a \times 10^n \]where \( 1 \fless a < 10 \)and \( n \)is an integer.
02

Place the Decimal

Position the decimal in the number \( 0.082 \) so that there is only one non-zero digit to its left. This gives \( 8.2 \).
03

Determine the Exponent

Count the number of places the decimal point has moved from the original number \( 0.082 \) to \( 8.2 \). It has moved 2 places to the right, so the exponent is \( -2 \).
04

Write in Scientific Notation

Combine the number from Step 2 with the exponent determined in Step 3: \[ 8.2 \times 10^{-2} \].

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Key Concepts

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

Decimal Placement
When converting a number into scientific notation, the first step is to correctly place the decimal point. You want to position the decimal such that there is only one non-zero digit to its left. For example, with the number 0.082, you would move the decimal point two places to the right, changing the number to 8.2. This is a critical step because it simplifies the number, making it easier to express as a multiplication of a digit and a power of ten.
Exponents
After placing the decimal correctly, we must determine the exponent for the power of ten. This exponent represents how many places the decimal point was moved to convert the original number to our new form. If we move the decimal to the right, as in 0.082 to 8.2, the exponent will be negative, because we are working with a number smaller than one. In our example, the decimal moved two places to the right, so the exponent is -2. This is because moving to the right decreases the value of the number, requiring a negative exponent to balance.
Expressing Numbers
Once you've determined the correct placement of the decimal and the necessary exponent, you can combine them to express the number in scientific notation. This form is useful for very large or small numbers, as it simplifies calculations and makes the numbers more readable. For instance, our converted number, 8.2, and the exponent, -2, combine to give us the scientific notation: \[8.2 \times 10^{-2}\]. This notation is compact and keeps the precision of the number while also showing the scale.

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Most popular questions from this chapter

Simplify the expression. $$\left(\frac{z}{2}\right)^{3}$$

Fill in the missing polynomial. $$(\quad)(z-5)=z^{2}-3 z-10$$

Perform the indicated operations. $$\left(-5 a^{2}-9 a^{3}+10 a^{4}\right)+\left(2 a^{3}-a^{4}\right)-\left(8 a+4 a^{3}+10 a^{4}\right)$$

Subtract the polynomials. $$\left(9 p^{3}-4 p^{2}+2 p\right)-\left(2 p^{3}+6 p^{2}-3 p\right)$$

Simplify. Write the answers with positive exponents only. $$\frac{x^{5}}{x^{-7}}$$
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