Powers and Negative Exponents: Rules & Examples

Exponent is any number raised on a base which can also be seen as a numerical force meaning that the number is multiplied by itself. It is represented by a² where a is base and 2 is its power. This can be read as a raise to power 2. Therefore, the method of writing large numbers in a short form using exponent is known as the exponential form. 

Keyterms: Power, Exponent, Factor, Base, Number, Exponential form.

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Powers and Exponents

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The expression that represents the repetition of the same factor is called power. For example, 5², which means 5 × 5 = 10. The number 5 is called the base, and the number 2 is called the power or exponent. The expectation corresponds to the number of times the base is used as a factor.

Powers and Exponents

Other examples, 4², 5⁸, 7⁷, 10³, etc. 

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Negative Exponents

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We know that positive exponents tell us how many times a number is multiplied by itself. Although, a negative exponent tells us how many times we have to multiply the reciprocal of the base number. In other words, a negative element explains how often we have to replicate the base. 

An example of negative exponents is 5^-4. 

Therefore, it can be represented as 1/5⁴.

After solving it we get 1/5 × 1/5 × 1/5 × 1/5 = 1/625.

More examples, 3^-4, 2^-6, 6^-3, 5^-7, 8^-2, etc. 

Note: If there is no exponent given in a number, its power is the same number which means it is its one time. For example, 2 - In this number, 1 is the power of 2 because 1 cannot be written in the power form.

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Laws of Exponent

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Some of the Laws of Exponents are described below.

Exponents with the same base 

• When two exponential numbers are multiplied with the same base - N⁴ × N³= N⁽⁴⁺³⁾
• When two exponential numbers are divided with the same base - N⁴/ N³ = N^(4-3)

Power of Powers

When a base has a power of power - (4⁴)², then the powers will be multiplied with each other. (4⁴)² = (4)^4x2 = 4⁸

Same Exponents but Different Bases

• When two numbers are multiplied with the same exponents - (4⁴ × 2⁴) = (4×2)⁴ = 8⁴
• When two numbers are divided with the same exponents - (4⁴ / 2⁴) = (4/2)⁴ = 2⁴

Also Read: Multiplication and Division of Integers

Use of Negative Exponents

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• Exponents are widely used in high-level calculations, which allow you to represent a number or variable that automatically repeats itself a certain number of times. For example, the expression 2⁴ represents a product of 2 × 2 × 2 × 2, which is equal to 16. 
• Some more applications include understanding the scientific scale as a pH scale or Richter scale, using a scientific note to write very large or very small numbers, and taking measurements.

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Things to Remember

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• The expression that represents the multiplication of the same number is called power.
• The negative exponent is defined as the multiplicative inverse of the base, raised to an exponent which is opposite the given exponent.
• a^-2 is also known as the multiplicative inverse of a².
• Powers are useful in expressing large expressions.
• A positive exponent determines how often we should multiply a base number, while a negative exponent determines how many times we should divide a base number.

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