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Chapter 5: Problem 80

Write each number in decimal notation without the use of exponents. $$2.75 \times 10^{3}$$

Short Answer

Expert verified
The number \(2.75 \times 10^{3}\) written in decimal notation is \(2750.0\).

Step by step solution

01

Identify the given number and the exponent

The given number is \(2.75\) and its exponent, which it is multiplied by, is \(10^{3}\).
02

Translate the exponent

Being multiplied by \(10^{3}\) means that the number needs to be shifted 3 places to the right. This is because an exponent of 3 on the base 10 means the number is multiplied by 10 three times, resulting in 3 zeros being added to or shifted in the number.
03

Rewrite the number in decimal notation

Shift the decimal point of \(2.75\) three places to the right. If there aren't enough digits to the right of decimal, add zeroes. Thus, \(2.75 \times 10^{3}\) becomes \(2750.0\).

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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 method used to express very large or very small numbers in a compact form. This system is especially useful in sciences and engineering to simplify recording and calculating with cumbersome figures.

Essentially, scientific notation follows the format of \( a \times 10^{n} \), where \( a \) is a number between 1 and 10, known as the coefficient, and \( n \) is an integer called the exponent. The exponent indicates how many places the decimal point has been moved. For example, the number 5300 in scientific notation is \( 5.3 \times 10^{3} \) because the decimal point is moved 3 places to the right of the original number.
Exponent
An exponent represents how many times a number, known as the base, is multiplied by itself. It is written as a small number to the upper-right of the base number. For instance, \(10^{3}\) tells is that the base, 10, is multiplied by itself three times: \(10 \times 10 \times 10\), which equals 1000.

Example:


If we have \(4^{2}\), it means \(4\) is squared, or \(4 \times 4\), resulting in 16. It's a shorthand way of writing multiplication that deals with the same number repeatedly.
Multiply by Powers of Ten
When you multiply a number by a power of ten, you're essentially shifting its decimal point to the right (for positive powers) or to the left (for negative powers) as many times as the exponent's value. Each power of ten multiplies the base number by 10.

Positive Powers:

  • \(\(10^{1} = 10\)\): Shift one place to the right.
  • \(\(10^{2} = 100\)\): Shift two places to the right.

Negative Powers:


For negative powers like \(10^{-1}\), it's the opposite; you shift the decimal point one place to the left, which is the same as dividing by ten.
Place Value
Place value in mathematics determines the value of a digit based on its position within a number. Each position in a number has a value that is a power of 10. Starting from the right, the first place is the ones, then tens, hundreds, thousands, and so on.

For example, in the number 1234, the digit 4 is in the ones place, 3 is in the tens place, 2 is in the hundreds place, and 1 is in the thousands place. This positional system is essential for understanding how to move the decimal point when multiplying by ten and its powers.

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Most popular questions from this chapter

Are the expressions $$ \frac{12 x^{2}+6 x}{3 x} \text { and } 4 x+2 $$ equal for every value of \(x ?\) Explain.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I wrote a number where there is no advantage to using scientific notation instead of decimal notation.

Graph \(3 x-2 y=6\) using intercepts. (Section 3.2 Example 4 )

In Exercises \(79-82,\) simplify each expression. $$\frac{2 x^{3}(4 x+2)-3 x^{2}(2 x-4)}{2 x^{2}}$$

In each exercise, find the product. $$(x+3)\left(x^{2}+5\right)$$
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