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What is an Exponent?

Finding the exponent of a number, or exponentiation is a mathematical operation. Exponentiation is defined as an where β€˜a’ is some numeric value, called the base, and n is the power or exponent.

If the base is e (Euler’s number), exponentiation is written as en. Calculating the exponent of a base is essentially repeated multiplication of the base n times.

For example, an = π‘Ž Γ— π‘Ž Γ— π‘Ž Γ— … π‘Ž 𝑛 π‘‘π‘–π‘šπ‘’π‘ . For the Euler’s number, en = 𝑒 Γ— 𝑒 Γ— 𝑒 Γ— … 𝑒 𝑛 π‘‘π‘–π‘šπ‘’π‘ . Therefore, if the base is 5 and the power is 3, 53 = 5 Γ— 5 Γ— 5 = 125.

Similarly, for the base e, e3 = 𝑒 Γ— 𝑒 Γ— 𝑒 = 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) = βˆ’125 and (βˆ’0.2) -3 = 1 / (-0.2x-0.2x-0.2) = 1 / 125 = 0.008

Properties of Exponent

Given below are some important properties of exponent.

Base – The number whose exponentiation is taking place. This can be any real number or Euler’s number, e.

Exponent – The exponent (or power) is the number to which the base is being raised. For example, for an, we say that the number β€˜a’ is being raised to the power β€˜n’. The exponent can be any real number.

Exponent Formula

The formula for exponent is

a) For any numeric base β€˜a’, an = π‘Ž Γ— π‘Ž Γ— π‘Ž Γ— … π‘Ž 𝑛 π‘‘π‘–π‘šπ‘’π‘ 

b) For Euler’s number, en = 𝑒 Γ— 𝑒 Γ— 𝑒 Γ— … 𝑒 𝑛 π‘‘π‘–π‘šπ‘’π‘ .

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

βˆ’an π‘Žπ‘›π‘‘ (βˆ’a)n give different answers. For example, βˆ’52 = βˆ’1 Γ— 5 Γ— 5 = βˆ’25, whereas, (βˆ’5)2 = (5) Γ— (βˆ’5) = 25. If a negative number is entered for the base, such as -5, the calculator automatically assumes (βˆ’5)n. When a negative sign is used with the exponent formula, (βˆ’a)n means that -a is raised to the power n. However, βˆ’an means the negative of β€˜a’ raised to the power β€˜n’.

Therefore, βˆ’an = βˆ’1 Γ— an = βˆ’1 Γ— π‘Ž Γ— π‘Ž Γ— … Γ— π‘Ž (𝑛 π‘‘π‘–π‘šπ‘’π‘ )

and (βˆ’a)n = (βˆ’π‘Ž) Γ— (βˆ’a) Γ— … Γ— (βˆ’a) (𝑛 π‘‘π‘–π‘šπ‘’π‘ )

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.

  • Exponents often come up in the case of measurements, when we use units like square feet, square inches, cubic feet, cubic inches, cubic metres and so on.
    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.

  • The same measurements are also used when we try to calculate the area of a square or the volume of a cube.
    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.

  • Negative exponents are used to indicate very small quantities, and large exponents are for enormous quantities.
    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 β€˜mega’ indicates 106, so one megabyte means one million bytes. The prefix β€˜kilo’ indicates 103, 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— 1024 kg. This is equal to 5.972 Γ— 10 Γ— 10 Γ— 10…. Γ— 10 (24 times).

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