Question

Express each of the following powers of 10 as an ordinary number (a) \(1 \times 10^{3}\) (b) \(1 \times 10^{-7}\)

Short answer

(a) 1000; (b) 0.0000001.

Step-by-step solution

Interpret the power of 10

The expression given is \(1 \times 10^3\). The power of 10 tells us how many places to shift the decimal point. A positive exponent indicates that we should move the decimal point to the right.

Shift the decimal to the right

Start with the number 1 and move the decimal three places to the right (since the exponent is 3), which gives us 1000.

Result for (a)

The expression \(1 \times 10^3\) as an ordinary number is 1000.

Interpret the power of 10 (for negative exponent)

The expression is now \(1 \times 10^{-7}\). The negative exponent means we need to move the decimal point to the left.

Shift the decimal to the left

Start with the number 1 and move the decimal seven places to the left. This involves adding zeros in front of the 1, resulting in 0.0000001.

Result for (b)

The expression \(1 \times 10^{-7}\) as an ordinary number is 0.0000001.

Key concepts

Powers of 10

Powers of 10 are a fundamental concept in scientific notation that allows us to express very large or very small numbers efficiently. They are written in the format of a number followed by a base of 10 raised to an exponent. In the expression \( a \times 10^n \),

• \(a\) is known as the coefficient.
• \(10^n\) is the power of 10.
• \(n\) is the exponent.

The exponent indicates how many times the base number 10 needs to be multiplied by itself. For example, \(10^3 = 10 \times 10 \times 10 = 1000\).
When dealing with powers of 10, recognizing the role of the exponent is key to easily and swiftly converting numbers in scientific notation back into their ordinary number form. This power to shift decimal points is what makes scientific notation such a powerful tool for working with numbers across many scientific and mathematical fields.

Positive Exponents

Positive exponents are straightforward to handle because they tell us to multiply the base number, which is usually 10 in our case, multiple times. These exponents are represented by positive numbers. For example, in the expression \(10^3\), the positive exponent 3 means we multiply the base 10 three times:

• First multiplication produces 10.
• Second multiplication results in 100.
• Third multiplication gives us 1000.

Similarly, the expression \(1 \times 10^3\) with a positive exponent of 3 shifts the decimal point three places to the right, turning 1 into 1000.
This movement of the decimal point to the right enlarges the number. If you think about it, using positive exponents is much like adding zeros to the end of the number.

Negative Exponents

Negative exponents work differently compared to positive exponents. Rather than multiplying, they tell us to divide or shift the decimal point to the left. In the expression \(1 \times 10^{-7}\), the negative exponent \(-7\) indicates that we should move the decimal point seven places to the left.
This transformation can be visualized by starting with 1 and inserting seven zeros in front of it, resulting in the tiny number 0.0000001.

• The first zero takes the place of a tenth.
• The second becomes a hundredth, and so on.
• By the seventh place, you have a tenth of a millionth.

In essence, a negative exponent indicates a division process where the number becomes a fraction of its original value, making it extremely small. Negative exponents are crucial for working with vast ranges of numbers, especially when approaching quantities that are much less than one.