## Calculate the Exponent value.

#### Exponent Calculator

Result:

b^{n} = : |
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### What is an Exponent?

Finding the exponent of a number, or exponentiation is a mathematical operation. Exponentiation is defined as a^{n} where βaβ is some numeric value, called the base, and n is the power or exponent.

If the base is e (Eulerβs number), exponentiation is written as e^{n}. Calculating the exponent of a base is essentially repeated multiplication of the base n times.

For example, a^{n} = π Γ π Γ π Γ β¦ π π π‘ππππ . For the Eulerβs number, e^{n} = π Γ π Γ π Γ β¦ π π π‘ππππ . Therefore, if the base is 5 and the power is 3, 5^{3} = 5 Γ 5 Γ 5 = 125.

Similarly, for the base e, e^{3} = π Γ π Γ π = 20.0855.

The exponent can be positive or negative real numbers.

For example, if the base is 5 and the power is -3, 5^{-3} = 1/ (5x5x5) = 1/125 = 0.008.

For the base e, e^{-3} = 1 / (exexe) = 1/(20.0855) = 0.04979.

Even the numeric base can be a positive or negative real number. For example, 0.2 ^{-3} = 1 / (0.2x0.2x0.2) = β125 and (β0.2) ^{-3} = 1 / (-0.2x-0.2x-0.2) = 1 / 125 = 0.008

### Properties of Exponent

Given below are some important properties of exponent.

**Base** β The number whose exponentiation is taking place. This can be any real number or Eulerβs number, e.

**Exponent** β The exponent (or power) is the number to which the base is being raised. For example, for a^{n}, we say that the number βaβ is being raised to the power βnβ. The exponent can be any real number.

### Exponent Formula

The formula for exponent is

a) For any numeric base βaβ, a^{n} = π Γ π Γ π Γ β¦ π π π‘ππππ

b) For Eulerβs number, e^{n} = π Γ π Γ π Γ β¦ π π π‘ππππ .

When calculating the exponent of a number, one must be careful of the negative signs.

βa^{n} πππ (βa)^{n} give different answers. For example, β5^{2} = β1 Γ 5 Γ 5 = β25, whereas, (β5)^{2} = (5) Γ (β5) = 25. If a negative number is entered for the base, such as -5, the calculator automatically assumes (β5)^{n}. When a negative sign is used with the exponent formula, (βa)^{n} means that -a is raised to
the power n. However, βa^{n} means the negative of βaβ raised to the power βnβ.

Therefore, βa^{n} = β1 Γ a^{n} = β1 Γ π Γ π Γ β¦ Γ π (π π‘ππππ )

and (βa)^{n} = (βπ) Γ (βa) Γ β¦ Γ (βa) (π π‘ππππ )

### Exponent rules

Exponentiation follows several rules which have been listed below.

rule | example |
---|---|

Product rule with the same base: $$ a^m \,* \, a^n = a^{m+n} $$ | For example: $$ 3^4 \,* \, 3^2 = 3^{4+2} \, = 3^6 \, = 729 $$ |

Product rule with the same exponent, but different base: $$ a^n \,* \, b^n = (a*b)^n $$ | For example: $$ 2^4 \,* \, 3^4 = (2*3)^4 \, = 6^4 \, = 1296 $$ |

Quotient rule with the same base: $$ {{a^m}\over{a^n}} = a^{m-n} $$ | For example: $$ {{3^4}\over{3^2}} = 3^{4-2} = 3^2 = 9 $$ |

Quotient rule with the same exponent, but different base: $$ {{a^n}\over{b^n}} = ({a \over b})^n $$ | For example: $$ {4^5 \over 2^5} = ({4 \over 2})^5 \, = 2^5 \, = 32 $$ |

Power rule 1: $$ (a^m)^n = a^{m*n} $$ | For example: $$ (4^2)^3 = 4^{(2*3)} = 4^6 = 4096 $$ |

Power rule 2: $$ {a^{m^n}} = a^{(m^n)} $$ | For example: $$ {2^{3^2}} = 2^{(3^2)} = 2^9 = 512 $$ |

Power rule with radicals: $$ n\sqrt{a^m} = a^{m\over n} $$ | For example: $$ 3\sqrt{4^6} = 4^{6\over3} = 4^2 = 16 $$ |

Negative exponent rule 1: $$ a^{-n} = {1 \over a^n} $$ | For example: $$ 2^{-3} = {1 \over 2^3} = {1\over8} $$ |

Negative exponent rule 2: $$ ({a\over b})^{-n} = {b^n \over a^n} $$ | For example: $$ ({6\over3})^{-2} = {3^2 \over 6^2} = {9\over36} = {1\over4} $$ |

### Areas of application

Exponentiation is the process of multiplying a number several times by itself. We use this frequently in daily life.

For example, one square inch = 1 inch Γ 1 inch = (1 inch)2. Similarly, one cubic inch = 1 inch Γ 1 inch Γ 1 inch = (1 inch)

^{3}.

For example, the area of a square floor of size 10 feet = 10 feet Γ 10 feet = (100 feet)

^{2}= 100 square feet.

Similarly, the volume of a cubic container, where each side measures 5 inches is, 5 inches Γ 5 inches Γ 5 inches = (125 inch)

^{3}= 125 cube inch.

For example, micrometre means 10

^{-6}metre. This is used in a device called screw gauge to measure the thickness of very thin objects like paper. On the contrary, in computer science and technology, we frequently come across terms like kilobyte, megabyte and gigabyte. The prefix βmegaβ indicates 10

^{6}, so one megabyte means one million bytes. The prefix βkiloβ indicates 10

^{3}, so one kilobyte means one thousand bytes.

To measure the mass of heavenly bodies like planets or moons, exponents are used again. For example, the mass of the Earth is 5.972 Γ 10

^{24}kg. This is equal to 5.972 Γ 10 Γ 10 Γ 10β¦. Γ 10 (24 times).