Exponents Study Guide: GED Math

Fractional Exponents

When an exponent is a fraction, the denominator of this fractional exponent means the root of the base number, and the numerator means a raise of the base to that power.

numerator—the power to which the base is raised

denominator—the root of the base number

For example, 8^1/3 is the same as = 2. Another example is 8^2/3 means , which is = 4, or alternatively, , which is 2² = 4.

Scientific Notation

Scientists measure very large numbers, such as the distance from the earth to the sun, or very small numbers, such as the diameter of an electron. Because these numbers involve a lot of digits as placeholders, a special notation was invented as a shorthand for these numbers. This scientific notation is very specific in the way it is expressed:

n × 10^e

where n = a number greater than 1 and less than 10

e = the exponent of 10

To find the scientific notation of a number greater than 10, locate the decimal point and move it either right or left so that there is only one non-zero digit to its left. The result will produce the n part of the standard scientific notational expression. Count the number of places that you had to move the decimal point. If it is moved to the left, as it will be for numbers greater than 10, that number of positions will equal the e part.

Example

What is 23,419 in scientific notation?

Position the decimal point so that there is only one non-zero digit to its left: 2.3419

Count the number of positions you had to move the decimal point to the left, and that will be e: 4.

In scientific notation, 23,419 is written as 2.3419 × 10⁴.

For numbers less than 1, follow the same steps except in order to position the decimal point with only one non-zero decimal to its left, you will have to move it to the right. The number of positions that you move it to the right will be equal to –e. In other words, there will be a negative exponent.

Example

What is 0.000436 in scientific notation?

First, move the decimal point to the right in order to satisfy the condition of having one non-zero digit to the left of the decimal point: 4.36

Then, count the number of positions that you had to move it: 4.

That will equal –e, so –e = –4, and 0.000436 = 4.36 × 10^–4.

For numbers already between 1 and 10, you do not need to move the decimal point, so the exponent will be zero.

Laws of Exponents

When working with exponents, there are three rules that can be helpful.

1. When you multiply powers with the same base, keep the base and add the exponents: 4² × 4³ = 4^{2 + 3} = 4⁵.
2. When you divide powers with the same base, keep the base and subtract the exponents: 3⁷ ÷ 3⁴ = 3^{7 – 4} = 3³.
3. When you raise a power to a power, you keep the base and multiply the exponents: (7³)² = 7^{3 × 2} = 7⁶.

If two powers are being multiplied together and the bases are not the same, check to see if you can convert the numbers to have the same base to use the preceding laws. For example, to simplify 27² × 3², recognize that 27 can be written as 3³. Change the problem to (3³)² × 3². Use law number 3 to get 3^{3 × 2} × 3² = 3⁶ × 3². Now, you can use law number 1 to get 3^{6 + 2} = 3⁸.

Exponents and the Order of Operations

Be aware of some distinctions when working with the order of operations and exponents. Exponents are done after parentheses and before any other operations, including the negative sign. For example, –3² = –(3 × 3) = –9 because you first take the second power of 3 and then the answer is negative. However, (–3)² = –3 × –3 = 9, because –3 is enclosed in parentheses.

Example

–20 + (–2 + 5)3 ÷ (10 – 7) × 2 =

Evaluate the parentheses, left to right:

–20 + 33 ÷ 3 × 2.

Now, evaluate the exponent: –20 + 27 ÷ 3 × 2.

Division will be done next: –20 + 9 × 2.

Evaluate multiplication: –20 + 18.

Finally, perform addition to arrive at the answer: –2.

Practice problems for these concepts can be found at:

Exponents and Roots Practice Problems: GED Math