Question

Express each of the following powers of 10 as an ordinary number: (a) \(1 \times 10^{12}\) (b) \(1 \times 10^{-2}\)

Short answer

(a) 1,000,000,000,000 (b) 0.01

Step-by-step solution

Understanding Positive Powers of 10

When you have a positive power of 10, it indicates that you should move the decimal point to the right. In the expression \(1 \times 10^{12}\), the exponent "12" tells us to move the decimal point 12 places to the right from its position in the number 1.

Writing an Ordinary Number for Positive Powers of 10

Starting with the number 1, which has an implied decimal point right after it (1.0), moving the decimal 12 places to the right leaves us with the number 1000000000000. Therefore, \(1 \times 10^{12} = 1,000,000,000,000\).

Understanding Negative Powers of 10

For a negative power of 10, this means moving the decimal point to the left. In the expression \(1 \times 10^{-2}\), the exponent "-2" tells us to move the decimal point 2 places to the left from its position in the number 1.

Writing an Ordinary Number for Negative Powers of 10

Starting with the number 1, which has an implied decimal point right after it (1.0), moving the decimal 2 places to the left renders the number as 0.01. Hence, \(1 \times 10^{-2} = 0.01\).

Key concepts

Powers of 10

Powers of 10 are a fundamental concept in mathematics, especially in scientific notation, which allows for the simplification of very large or very small numbers. It involves multiplying the number 10 raised to an exponent. The exponent indicates how many times the number 10 is multiplied by itself. For example:

• In the expression \(10^3\), the "3" tells us to multiply 10 three times: \(10 \times 10 \times 10 = 1000\).
• Conversely, in \(10^{-3}\), the negative "-3" signifies division instead of multiplication, representing \(\frac{1}{10^3}\) or \(\frac{1}{1000} = 0.001\).

The utility of powers of 10 becomes evident in scientific fields, such as physics and chemistry, where they help represent very large distances or very tiny particles. When combined with positive or negative exponents, they make calculations straightforward and reduce computational errors.

Positive Exponents

Positive exponents are used in powers of 10 to denote large numbers. They describe how many times the base number, which is 10 in this case, needs to be multiplied by itself. When you see a positive exponent, it means the decimal point in the given number shifts to the right for each step up the exponent.Let's take the example \(1 \times 10^{12}\):

• The positive exponent "12" indicates moving the decimal point 12 times to the right.
• Start with 1, or as an implied decimal, 1.0.
• After shifting 12 places to the right, you turn this into 1,000,000,000,000 – a trillion!

Using positive exponents in scientific notation simplifies expressing large figures, making it convenient for calculations and communication.

Negative Exponents

Negative exponents represent fractional numbers (numbers less than one) in powers of 10. These are helpful when you need to work with very small numbers. Instead of multiplying the number, negative exponents indicate division.Here's how it works in the example of \(1 \times 10^{-2}\):

• The negative exponent "-2" suggests moving the decimal point to the left.
• Beginning with 1.0, two decimal places to the left transforms it into 0.01.

The expression \(10^{-2}\) equates to \(\frac{1}{10^2}\), or 0.01 in decimal form. Negative exponents simplify the handling of small measurements, vital especially in fields like chemistry and biology where microscopic scales are standard. This helps in avoiding lengthy strings of zeros and ensures precision in scientific communication and data handling.