Powers with negative exponent and negative base. Resolved exercises.

Do you have problems solving exercises with NEGATIVE POWERS? Negative powers can take two forms: negative exponent powers or negative base powers.

Next I will explain step by step how you have to operate with the negative exponent powers. We will also see how to operate with the powers that have a negative base.

In short, I will teach you everything you need to know about the minus signs in the powers, whether they are at the base or in the exponent.

Formula for converting powers from negative exponent to positive exponent

There is a power property for negative exponent powers which says that a power with a negative exponent is equal to 1 between the same power, but with a positive exponent.

In other words, to convert the exponent from negative to positive, the power is converted to the denominator:

potencias de exponente negativo

For example:

base negativa y exponente negativo

When you operate with numbers, it is not possible to obtain a direct result with the powers of negative exponent, so in order to reach its result, first you must convert the exponent to positive by applying the previous formula:

exponente negativo

A particular case of power with negative exponent is the inverse of a number. The inverse of a number, is that number raised to -1, and is resolved in this way:

potencias negativas

Fractions with negative exponent

When we have fractions elevated to a negative exponent, to convert the exponent to positive, we must turn the fraction around, according to the following formula:

potencias con exponente negativo

In reality, when the base is not a fraction, i. e. it is a number or a variable, we also turn it upside down, since we place the power in the denominator and the original denominator (which was a 1) becomes the numerator.

For example:

potencia de exponente negativo

First we turn the fraction and the exponent to positive and then eliminate parentheses by applying the properties of the powers.

Let’s see another example fractions with numbers:

potencia con exponente negativo

Once we have passed the exponent to positive, we can already get a result of the power.

Now that you’ve learned to work with negative exponents, it’s time for the powers that have the minus sign at the base.

Negative Base Powers

Powers can have a minus sign in the base, but be careful because you have to differentiate 2 cases:

1 – The minus sign is affected by the exponent. We know that the minus sign is affected by the exponent because the number or variable and the minus sign are enclosed in parentheses.

Then you have to keep in mind that:

If the exponent is even, the result will be positive:

exponentes negativos ejemplos

If the exponent is odd, the result will be negative:

base y exponente negativo

2 – The minus sign is NOT affected by the exponent. The minus sign does NOT belong to power and is totally independent.

How do we know if the minus sign is affected or not by the exponent? Because he’s not enclosed in parentheses.

potencia negativa

In other words, the minus sign is added to the result of the power (irrespective of whether it is positive or negative).

Negative Base Powers examples

Let’s start to solve some potentials so that you have a clear idea of how to deal with the negative signs at the base.

Example 1:

potencias negativas con exponente negativo

The minus sign is affected by the exponent because it is enclosed in parentheses. As the exponent is odd, the result is negative:

ejemplos de exponente negativo

Example 2:
fracciones con exponente negativo

The minus sign is not affected by the exponent. But even so, the result is negative because we already had the minus sign and it is added to the power result.

The result does not depend on whether the exponent is even or odd.

potencias exponente negativo

Example 3:

exponentes negativos en fracciones

As in the previous example, the minus sign is not affected by power and does not depend on whether the exponent is even or odd. We just put the minus sign, which we already had and solve the potency

potencia de base negativa

Example 4:

denominador con exponente negativo

Again, the minus sign belongs to the potency. The exponent is even, then the result is positive:

operaciones con exponentes negativos

If you notice, you just have to pay close attention if the minus sign is in parentheses or not to know how to act.

Now let’s solve some examples by mixing the negative base powers and the negative exponent powers.

Solved exercises of negative exponent and negative base powers.

How to operate when we have fewer signs in the base and in the exponent?

We are going to solve some examples step by step that will make things much clearer:

Example 1:

potencia exponente negativo

To begin with, the minus sign on the base does not belong to power. Therefore, the result has a minus sign that we already had and the result of applying the property of the negative exponent, passing the power to the denominator:

fracciones con potencias negativas

Example 2:

potenciación con exponente negativo

The minus sign belongs to the power, therefore, the whole base (with the minus sign included) passes to the denominator:

ejemplos de potencias con exponente negativo

And now we solve the power with a positive exponent that we have in the denominator. We have a negative basis, with an odd exponent, so the result is negative:

potenciación de de exponente negativo

Example 3:

potencia negativa en fracciones

This exercise is a bit more complicated, as we have the power with negative exponent in the denominator.

Just as when we have the negative exponent in the numerator and goes to the denominator, when the negative exponent is in the denominator, it goes to the numerator with positive exponent:

exponentes negativos

Therefore, the power of the example would be like this: