# Powers with negative integer exponent - Examples, Exercises and Solutions

When we see any number(positive or negative) raised to a negative power we can convert the expression into a fraction and we will do it as follows:
the numerator will be $1$, the denominator will be the base of the power as seen in the original exercise, but now, with a positive exponent.
That is to say, in the denominator we will invert the exponent to positive.
Pay attention, we will not modify the sign of the base of the potentiation even if it is negative.
Property formula:
$a^{-n}=\frac {1}{a^n}$
This property also applies to algebraic expressions.

## Practice Powers with negative integer exponent

### Exercise #1

$(\frac{1}{4})^{-1}$

### Step-by-Step Solution

We use the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We will write the fraction in parentheses as a negative power with the help of the previously mentioned power:

$\frac{1}{4}=\frac{1}{4^1}=4^{-1}$We return to the problem, where we obtained:

$\big(\frac{1}{4}\big)^{-1}=(4^{-1})^{-1}$We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$And we apply it in the problem:

$(4^{-1})^{-1}=4^{-1\cdot-1}=4^1=4$Therefore, the correct answer is option d.

$4$

### Exercise #2

$5^{-2}$

### Step-by-Step Solution

We use the property of powers of a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the problem:

$5^{-2}=\frac{1}{5^2}=\frac{1}{25}$

Therefore, the correct answer is option d.

$\frac{1}{25}$

### Exercise #3

$4^{-1}=\text{?}$

### Step-by-Step Solution

We use the property of raising to a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the problem:

$4^{-1}=\frac{1}{4^1}=\frac{1}{4}$Therefore, the correct answer is option B.

$\frac{1}{4}$

### Exercise #4

$7^{-24}=\text{?}$

### Step-by-Step Solution

We use the property of raising to a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it in the problem:

$7^{-24}=\frac{1}{7^{24}}$Therefore, the correct answer is option D.

$\frac{1}{7^{24}}$

### Exercise #5

$19^{-2}=\text{?}$

### Step-by-Step Solution

To solve the exercise, we use the property of raising to a negative exponent

$a^{-n}=\frac{1}{a^n}$

We use the property to solve the exercise:

$19^{-2}=\frac{1}{19^2}$

We can continue and solve the power

$\frac{1}{19^2}=\frac{1}{361}$

$\frac{1}{361}$

### Exercise #1

$\frac{1}{8^3}=\text{?}$

### Step-by-Step Solution

We use the power property for a negative exponent:

$b^{-n}=\frac{1}{b^n}$We apply it to the problem:

$\frac{1}{8^3}=8^{-3}$When we use this previously mentioned property in the opposite sense.

Therefore, the correct answer is option A.

$8^{-3}$

### Exercise #2

$\frac{1}{2^9}=\text{?}$

### Step-by-Step Solution

We use the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the expression we obtained:

$\frac{1}{2^9}=2^{-9}$

Therefore, the correct answer is option A.

$2^{-9}$

### Exercise #3

$\frac{1}{12^3}=\text{?}$

### Step-by-Step Solution

First, we recall the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the expression we obtained:

$\frac{1}{12^3}=12^{-3}$Therefore, the correct answer is option A.

$12^{-3}$

### Exercise #4

$[(\frac{1}{7})^{-1}]^4=$

### Step-by-Step Solution

We use the power property of a negative exponent:

$a^{-n}=\frac{1}{a^n}$We will rewrite the fraction in parentheses as a negative power:

$\frac{1}{7}=7^{-1}$Let's return to the problem, where we had:

$\bigg( \big( \frac{1}{7}\big)^{-1}\bigg)^4=\big((7^{-1})^{-1} \big)^4$We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$And we apply it in the problem:

$\big((7^{-1})^{-1} \big)^4 =(7^{-1\cdot-1})^4=(7^1)^4=7^{1\cdot4}=7^4$Therefore, the correct answer is option c

$7^4$

### Exercise #5

$2^{-5}=\text{?}$

### Step-by-Step Solution

We use the property of raising to a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the problem:

$2^{-5}=\frac{1}{2^5}=\frac{1}{32}$Therefore, the correct answer is option A.

$\frac{1}{32}$

### Exercise #1

$(-7)^{-3}=\text{?}$

### Step-by-Step Solution

We use the power property for a negative exponent:

$b^{-n}=\frac{1}{b^n}$We apply it in the problem:

$(-7)^{-3}=\frac{1}{(-7)^3}$When we notice that each whole number in parentheses is raised to a negative power (that is, the number and its negative coefficient together), by using the previously mentioned power property we were careful and took this fact into account,

We continue simplifying the expression in the denominator of the fraction, remembering the power property for the power of terms in multiplication:

$(a^m)^n=a^{m\cdot n}$We apply the expression we obtained:

$\frac{1}{(-7)^3}=\frac{1}{(-1\cdot7)^3}=\frac{1}{(-1)^3\cdot7^3}=\frac{1}{-1\cdot7^3}=\frac{1}{-7^3}=-\frac{1}{7^3}$

Summarizing the solution to the problem, we obtained that:

$(-7)^{-3}=\frac{1}{(-7)^3}=\frac{1}{-7^3}=-\frac{1}{7^3}$

Therefore, the correct answer is option B.

$-\frac{1}{7^{3}}$

### Exercise #2

$a^{-4}=\text{?}$

$(a\ne0)$

### Step-by-Step Solution

We use the property of raising to a negative exponent:

$b^{-n}=\frac{1}{b^n}$We apply it in the problem:

$a^{-4}=\frac{1}{a^4}$Therefore, the correct answer is option B.

$\frac{1}{a^4}$

### Exercise #3

$\frac{1}{(-2)^7}=?$

### Step-by-Step Solution

First, we deal with the expression in the denominator of the fraction and remember according to the property of raising an exponent to another exponent:

$(a^m)^n=a^{m\cdot n}$We obtain that:

$(-2)^7=(-1\cdot2)^7=(-1)^7\cdot2^7=-1\cdot2^7=-2^7$We return to the problem and apply what was said before:

$\frac{1}{(-2)^7}=\frac{1}{-2^7}=\frac{1}{-1}\cdot\frac{1}{2^7}=-\frac{1}{2^7}$When in the last step we remember that:

$\frac{1}{-1}=-1$Next, we remember the property of raising to a negative power

$a^{-n}=\frac{1}{a^n}$We apply it to the expression we obtained in the last step:

$-\frac{1}{2^7}=-2^{-7}$Let's summarize the steps of the solution:

$\frac{1}{(-2)^7}=-\frac{1}{2^7} = -2^{-7}$

Therefore, the correct answer is option C.

$(-2)^{-7}$

### Exercise #4

$x^{-a}=\text{?}$

### Step-by-Step Solution

We use the property of raising to a negative exponent:

$b^{-n}=\frac{1}{b^n}$We apply it in the problem:

$x^{-a}=\frac{1}{x^a}$Therefore, the correct answer is option C.

$\frac{1}{x^a}$

### Exercise #5

$7^{-4}=\text{?}$

### Step-by-Step Solution

First, we recall the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the expression we obtained:

$7^{-4}=\frac{1}{7^4}=\frac{1}{2401}$

Therefore, the correct answer is option C.

$\frac{1}{2401}$