Power with negative exponent: how to calculate, examples and exercises

Negative exponent power is a mathematical operation where a base is raised to an integer exponent less than zero.

Example
5 to the power of minus 2 end of the exponential
Where the exponent is -2 and the base is five.

In a power, the base is multiplied by itself as many times as the value of the exponent indicates.

Example
2 cubed equals 2 multiplication sign 2 multiplication sign 2 equals 8
Where 2 is the base, 3 is the exponent, and 8 is the result or power.

In case the exponent is negative, we have two situations: fractional base and integer base.

Fractional base raised to a negative exponent

A fraction raised to a negative exponent is inverted, the numerator becomes the denominator, and the denominator goes up, going to the numerator. Afterwards, the fraction is raised to the same exponent, this time positive.

Example
open parentheses 2 over 3 close parentheses to the power of minus 2 end of exponential equals open parentheses 3 over 2 closes parentheses squared equals 3 squared over 2 squared equals numerator 3 space. space 3 over denominator 2 space. space 2 end of fraction equals 9 over 4

Integer base raised to a negative exponent

Every integer can be written as a fraction with a denominator of 1, since every number divided by 1 results in itself.

Example
4 to the power of minus 2 end of the exponential equals open parentheses 4 over 1 closes parentheses to the power of minus 2 end of the exponential

So, just proceed as in the previous case, inverting the fraction and raising it to the modulus of the exponent, that is, the same numerical value, now positive.

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4 to the power of minus 2 end of the exponential equals open parenthesis 4 over 1 close parenthesis to the power of minus 2 end of exponential equals open parentheses 1 quarter closes parentheses squared equals 1 squared over 4 squared equals numerator 1 space. space 1 over denominator 4 space. space 4 end of fraction equals 1 over 16

Rule of thumb for integer base and negative exponent

The power goes to the denominator of a fraction with numerator 1, already with a positive exponent.

Example
4 to the power of negative 2 end of the exponential equals 1 over 4 squared equals 1 over 16

Power exercises with negative exponents

Exercise 1

calculate the power 5 to the power of minus 3 end of the exponential.

5 to the power of minus 3 end of exponential equals 1 over 5 cubed equals numerator 1 over denominator 5 multiplication sign 5 multiplication sign 5 end of fraction equals 1 over 125

Exercise 2

solve 2 to the minus 3 power end of the exponential space multiplication sign space opens parentheses 6 over 7 closes parentheses to the minus 2 power end of the exponential.

2 to the power of minus 3 end of exponential space multiplication sign space opens parentheses 6 over 7 closes parentheses to the power of minus 2 end of exponential equals 1 over 2 cubed multiplication sign opens parentheses 7 over 6 closes squared parentheses equals 1 over 2 cubed multiplication sign 7 squared over 6 squared equals 1 over 8 multiplication sign 49 over 36 equals 49 over 288

See too

  • Potentiation
  • Strengthening Exercises
  • Potentiation properties
  • Powers of base 10
  • perfect square
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