Question

Express each of the following ordinary numbers as a power of 10: (a) 100,000,000,000,000,000 (b) 0.000000000000001

Short answer

(a) \(10^{17}\); (b) \(10^{-15}\).

Step-by-step solution

Understanding Powers of 10

The power or exponent of 10 indicates the number of zeros following a 1 for positive powers, or the number of times you move the decimal to the left for negative powers. Let's apply this concept to both parts of the problem.

Identify the Number of Zeros for Part A

The number 100,000,000,000,000,000 has 17 zeros. Since there are no additional non-zero digits, this number can be expressed as 10 raised to the power of the number of zeros.

Express Part A as a Power of 10

Given the 17 zeros in the number, 100,000,000,000,000,000 can be written as a power of ten with an exponent representing these zeros. Thus, it is expressed as \(10^{17}\).

Identify Decimal Points for Part B

The number 0.000000000000001 has 15 decimal places before the first non-zero digit. This will determine the negative exponent when expressing as a power of 10.

Express Part B as a Power of 10

To express 0.000000000000001 as a power of ten, count the decimal places (which are 15 here) to the left of the first 1. This can be expressed as \(10^{-15}\).

Key concepts

Exponents

Exponents, sometimes known as powers, are mathematical notations used to indicate the number of times a number is multiplied by itself. For example, in the expression \(10^3\), the number 10 is the base and 3 is the exponent. This means "10 multiplied by itself 3 times" or \(10 \times 10 \times 10\). The resulting product is 1000.

Exponents make it easier to write and work with large numbers without having to write them out in full. Instead of writing 100,000, we can express it as \(10^5\), indicating that 10 is multiplied by itself five times.

• Base: The number that is repeatedly multiplied.
• Exponent: Tells how many times to multiply the base by itself.
• Expression: The entire notation, such as \(10^5\).

Positive Powers

When dealing with positive powers of 10, we are essentially determining how many zeroes follow the number 1. A positive exponent means the base number 10 is multiplied by itself a specific number of times. For example, \(10^4\) results in 10,000, displaying four zeros after the 1.

Here's an easy method:

• Each increase in the exponent represents an additional zero.
• \(10^1\) is 10 (one zero), \(10^2\) is 100 (two zeros), \(10^3\) is 1,000 (three zeros), and so on.

Understanding positive powers of 10 is helpful for recognizing and working with large numbers effortlessly in scientific computations.

Negative Powers

Negative powers are the flip side of positive powers but follow an intuitive logic. Rather than adding zeros, a negative exponent indicates how many times to divide 1 by the base, which shrinks the number towards zero. For example, \(10^{-1}\) equals 0.1, or one tenth.

When determining the negative power, count how many decimal places precede a non-zero digit in a number less than one, such as 0.0001:

• For this example, move the decimal four places to write it as \(10^{-4}\).
• \(10^{-2}\) is 0.01, \(10^{-3}\) is 0.001, indicating two and three places respectively.

Negative exponents are valuable for small, fractional numbers often used in scientific notation, making them more compact and easier to manage.

Decimal Representation

Decimal representation involves expressing numbers in the form of decimals, where we use a dot to separate the whole number part from the fractional. For instance, 0.75 means seventy-five hundredths.

In terms of powers of 10, decimals benefit greatly from exponents to easily convey very small numbers. If you have a number like 0.00001, it's cumbersome to count each zero. Using negative powers of 10 provides a tidy alternative: \(10^{-5}\).

• This simplifies decimal numbers by converting between standard format and exponent format seamlessly.
• Learn how to switch between \(0.001\) to \(10^{-3}\) effortlessly by practicing these conversions.

The combination of decimals and exponents is essential for expressing figures practically and succinctly in fields like science and engineering.