Scientific Notation
Powers And Exponents
Scientific notation is most often used as a concise way of writing very large and small numbers. It involves writing a number between 1 and 10 multiplied by a power of 10. Any number can be written in scientific notation.
Translate numbers written in scientific notation into standard form by reading the exponent.
A number multiplied by itself can be written as follows:
2 X 2 X 2 = 2³ = 8.
In the expression 2³, the small superscript 3 is called the exponent, and the 2 is called the base. An exponent is a number that indicates repeated multiplication. It shows how many times to multiply the base, the number that is being multiplied by itself. A number that can be expressed with a base and an exponent is called a power. For example, 2³ is read as "2 to the 3rd power."
A negative exponent can be written with the base of the denominator in a fraction where the numerator is 1. The exponent then becomes positive.
5ー౩ means 1/5³ = 1/125 = 0.008
10-² means 1/10² = 1/100 = 0.01
It is possible to multiply and divide numbers with positive and negative exponents that have the same base.
Scientific Notation is a neat and quick way of writing
(i) very large or
(ii) very small numbers
a × 10 b
Example: 2.3 × 106
Scientific Notation to Standard Form
Exponent is positive
2.3 × 10 6 = 2,300,000
Exponent is negative
2.3 × 10 6 = 0.0000023
Proxima Centauri, the closest star to our own - the Sun - is just 40,208,000,000,000 km away.
Considering that a light year corresponds to a distance of 9 × 10 km,
which of the two distances is larger?
First, let’s review the terminology:
Base 2, exponent 5
Two is the base. Five is the exponent. The whole thing (25) is called a power.
Now we’ll start with the basic rules.
Roots
The opposite operation of raising a number to a power is to take its root. Square roots are the root with which we are most familiar. Taking the square root is essentially removing the square from the number.
Example
62 = 36
√36 = 6
(Technically, √36 can also be (-6), since (-6)2 also equals 36.)
To find roots other than the square root, follow the same process. Without a calculator, you can really only find perfect roots. You do this by working backward and using mental math. To find the √36, you have to think, that some number times itself equals 36. Algebraically, it would look like:
x • x = 36
For a cube root, it is the same, only there are three Xs instead of two.
Example
Example 1
For higher-numbered roots, the number of Xs is the same as the number of the root.
Example
Example 2
Generally, you will only be asked to do fairly simple problems on this topic; it’s better suited for a calculator than for mental math.
Fractional Powers:
Finding fractional powers is like finding roots. Fractional powers require a bit of logical thinking, but, for our purposes, we can examine some problems that only require some brainpower.
Briefly, here is the theory behind fractional powers. Let’s look at it with the help of an example.
161/2 = x
161/2 • 161/2 = 161/2+1/2 = 161
Utilize the rule of multiplying powers with the same base.
Now go back and replace 161/2 with x.
x • x = 161
So x = 4, which means that 161/2 = 4. That is the same as the square root of 16.
How about 491/2?
You should have said 7.
Other fractional powers with a numerator of 1 work the same way, so a power of 1/3 is just like finding the cube root of the given number.
Example
Example:
If the numerator is a number other than 1, it tells you how many of the root numbers you need. If we raise 27 to the 2/3 power, we still find the cube root, which is 3, but we need it twice. Multiply 3 x 3 and you get 9.
Example:
323/5 = ?
32 to the 1/5 power = 2
2 • 2 • 2 = 23 = 8
So, 323/5 = 8
Scientific notation is especially useful for expressing very large numbers (think astronomy) and very small numbers (think microbiology).
Numbers written in scientific notation are composed of two parts.
The first part is a number between 1 and 10 (it doesn’t have to be a whole number, many are decimal numbers).
There can only be one digit to the left of the decimal, so the largest number is really 9.999…. not 10.
The second part is 10 raised to a power, which can be negative or positive.
The two parts are multiplied together so that they look like these examples:
Convert from Scientific Notation to Standard Form:
To write a number given in scientific notation as a number in standard form, you move the decimal point. The basic numbers of the notation do not change; the decimal moves, and you may need some placeholder zeroes.
The number of places the decimal moves is equal to the exponent.
For a positive exponent, move the decimal to the right—you are making a big number.
For a negative exponent, move the decimal to the left—you are making a small number (But remember: You are not making a negative number.) Multiplying by 10−3 is the same as dividing by 103.
Examples:
3.24 x 105 = 324,000 (move the decimal 5 places to the right, filling in the 3 extra places with zeroes)
9.7021 x 10−3 = 0.0097021 (move the decimal 3 places to the left, filling in the 2 extra places with zeroes)
Now it is your turn to practice.
Convert from Standard Form to Scientific Notation:
To write a number given in standard form as a number in scientific notation, there are two steps.
First, write the decimal number. Remember, you can only have one place to the left of the decimal. You can have as many places to the right as necessary. Drop any placeholder zeroes.
Examples
517,000 becomes 5.17; 0.0004906 becomes 4.906
Second, raise the 10 to the correct power. To determine the power, count how many places you need to move the decimal to make your decimal number. The exponent will be positive if the standard form is a big number (greater than 1). The exponent is negative if the standard form of the number is between 0 and 1.
Examples
517,000 = 5.17 x 105 (the decimal has to move 5 places)
0.0004906 = 4.906 x 104 (the decimal has to move 4 places)
Review:
There are several rules for working with exponents and powers.
Most importantly:
To multiply powers with the same base, add their exponents.
To divide powers with the same base, subtract their exponents.
To raise a power to a power, multiply the exponents.
Any number raised to zero power is one.
The opposite operation of raising a number to a power is taking its root.
Finding the fractional powers of a number is the same as finding the root of a number:
Raising a number to the 1/2 power is the same as finding its square root.
Numbers in scientific notation are composed of a decimal number
between 1 and 10 and a power of 10.
Numbers between 0 and 1 have negative exponents.
Numbers greater than 1 have positive exponents.
