Problem 60 - Algebra and Trigonometry

Question

Write each number in decimal notation. $$ 2.721 \times 10^{8} $$

Step-by-Step Solution

Answer

272,100,000

Step 1: Understanding Scientific Notation

The number given is in scientific notation: \(2.721 \times 10^8\). This means we multiply 2.721 by 10 raised to the power of 8.

Step 2: Move the Decimal Point

To convert from scientific notation to decimal notation, move the decimal point to the right as many places as the exponent indicates. Here, the exponent is 8, so move the decimal point 8 places to the right.

Step 3: Sliding the Decimal

Start with 2.721. Move the decimal point 8 places to the right. You will move past the '7' and the '2', arriving at 272,100,000. If necessary, add zeroes until the decimal point has moved the full 8 places.

Key Concepts

Decimal Notation

Decimal notation is the standard way of writing numbers. It's the number form we're most familiar with in everyday life. Each digit in a decimal number has a place value, based on powers of ten. For example, in the number 2,356:

• The '2' is in the thousands place.
• The '3' is in the hundreds place.
• The '5' is in the tens place.
• The '6' is in the units or ones place.

Combining all these values gives us the complete number. Decimal notation can also include a decimal point, followed by digits that represent fractions. For instance, in 27.21, the digits after the decimal point represent tenths and hundredths. This clear, base-10 approach makes decimal notation straightforward and easy to manage for calculations.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of a number by itself. When you see a number written in the form of a base, like 10, followed by a small superscript number, this is an exponent. It tells you to multiply the base number by itself a specific number of times. For instance, in the expression \(10^8\):

• The "10" is the base.
• The "8" is the exponent.

This notation means you multiply 10 by itself 8 times: \(10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10\), simplifying to 100,000,000. Exponents help simplify expressions and make calculations with very large or very small numbers more manageable. Understanding how to work with exponents is crucial when converting numbers from scientific notation to decimal notation, as seen in the exercise where \(10^8\) significantly impacts the number's size.

Converting Scientific Notation

Converting numbers from scientific notation to decimal notation involves understanding the role of the exponent. Scientific notation expresses numbers as \(a \times 10^n\), where 'a' is a number, typically between 1 and 10, and 'n' is an integer indicating how many places to move the decimal point.
In the exercise example of \(2.721 \times 10^8\):

• The decimal point starts after 2.721.
• Move the decimal point 8 places to the right, as indicated by the exponent 8.
• If needed, fill any empty places with zeros.

Starting at 2.721, moving the decimal point 8 places to the right gives you 272,100,000. This conversion involves a straightforward understanding of the movement dictated by the exponent, transforming scientific notation into easily understandable decimal notation.