Exponent
Content
What is an exponent or potentiation of a number?
The exponent or power is a mathematical operation that indicates how many times we must multiply a number by itself. Among the exponents that stand out are 2 and 3, which have a special name, 2 as the second power or square of a number and is the result of taking it as a factor twice, 3 as the third power or cube of a number and is the result of taking it as a factor three times.
Learn more about: “Multiplication”. →
Parts of the exponent or potentiation
The potentiation consists of two basic parts:
- Base (b): It is the factor that is repeated.
- Exponent (n): Indicates the amount the base repeats.
It is represented as follows:
b^{n}
Assuming you have 2^{4}, the base would be 2 and the exponent would be 4, b =2 y n = 4.
Laws of the exponents
The following laws are basic and important concepts regarding the theory:
- Any number or quantity raised to zero power equals 1, this means that if the exponent is equal to zero the result will always be 1. For example: 2^{0} = 1, 3^{0} = 1, 6^{0} = 1
- Any number or quantity raised to the first power is equivalent to the base number, therefore, if the exponent is equal to one the result will always be the base number. For example: 2^{1} = 2, 3^{1} = 3, 6^{1} = 6
- If the base is 1 ( base = 1 ), the result always has a value of 1 regardless of the value of the exponent.For example: 1^{4}= 1, 1^{20} = 1, 1^{33}=1
- If the base is greater than 1 ( base > 1 ), the greater the exponent, the greater the result. For example: 2^{2} = 4, 2^{3} = 8, 2^{4} = 16
- If the base is less than 1 ( base < 1 ), the greater the exponent, the smaller the result. For example: 0.5^{2} = 0.25, 0.5^{3} = 0.125, 0.5^{4} = 0.0625
The following laws apply with respect to operations:
- Equal base: When multiplying potentiation of the same base the exponents are added, therefore, b^{n} x b^{m} = b^{(n + m)} , assuming that a = 2, n = 3 and m = 4 and substituting you have to 2^{3} x 2^{4} = 2^{(3+4)}. It would have (2x2x2) x (2x2x2x2) = (2x2x2x2x2x2x2), and thus it is verified that the same result is obtained.
- Same base: By dividing potentiation of the same base the exponents are subtracted, therefore, b^{n} ÷ b^{m} = b^{(n - m)}, assuming that a = 3, n = 4 and m = 2 and substituting you have to 3^{4} ÷ 3^{2} = 3^{(4 - 2)}. It would have (3x3x3x3) ÷ (3x3) = (3x3).
- Multiplication of exponents: When multiplying the exponents we increase the times that we should consider the base number, it is expressed as (b^{n})^{m}. The first thing to do is multiply n x m and the result is the amount the base repeats. Assuming that b = 2, n = 3 and m = 2 and substituting you have to (2^{3})^{2} = 2^{(2 x 3)}, it would result (2x2x2x2x2x2) = 2^{6}.
Note: It is important to consider the sign of the exponents.
The following table shows the formulas commonly used in exponents with respect to the laws of the exponents, for some they were not mentioned since it is possible to obtain them by means of the mentioned ones.
b^{1} = b |
b^{0} = 1 |
b^{-n} = 1 / (b^{n}) |
b^{n} x b^{m} = b^{(n + m)} |
b^{n} / b^{m} = b^{(n - m)} |
(b^{n})^{m} = b^{(n x m)} |
(ab)^{n} = a^{n} b^{n} |
(a / b)^{n} = a^{n} / b^{n} |
Negative exponent
To calculate the negative exponent we must divide 1 by the base value with its positive power, in other words it is the reciprocal, since in this way the positive exponent is had. In mathematical form it is expressed as follows:
b^{-1} = 1 / b^{n}
There is another way to express a negative exponent, since a negative exponent means how many times it is divided by the number, but it can be confusing and time consuming for large quantities. For example: 2^{-3}= 1÷2÷2÷2
Examples:
Exercise:
8^{-2} = ?
1^{-1} = ?
2^{-0} = ?
Exponent Technique
Something practical and functional is to start with the number 1 and then multiply or divide the times indicated by the exponent, it is a way not to forget that every number with zero exponent is equal to 1.
3^{3} |
1 x 3 x 3 x 3 |
27 |
---|---|---|
3^{2} |
1 x 3 x 3 |
9 |
3^{1} |
1 x 3 |
3 |
3^{0} |
1 |
1 |
3^{-1} |
1 / 3 |
0.333... |
3^{-2} |
1 / 3 / 3 |
0.111... |
3^{-3} |
1 / 3 /3 / 3 |
0.037... |
Fractional exponent
A fractional exponent corresponds to the root of a number, therefore; If the expose is expressed as a fraction, to perform the mathematical operation, it is convenient to transform into a root form, for example:
2^{2/4 = }^{4}√2^{2}
4^{3/5 = }^{5}√4^{3}
Learn more about: “Root of a number”. →
Content
Arithmetic Tutorials
- Arithmetic
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