Negative numbers are the numbers less than zero and they appear to left side of zero on the number line. Exponent can be defined as the power to which a number is being raised. It can also be explained as how many times a number is multiplied by itself. For example if it is given $3^{2}$ it means 3 is multiplied by 3 once. If it is given that $3^{3}$ then it implies that 3 is multiplied by 3 twice, that is $3\times 3\times 3$. There can be exponents of negative numbers just like there are exponent for positive numbers. Exponent of negative number will multiply the number by itself the number of times as the power is being given.

## Negative Numbers with Exponents Definition

The exponent of a negative number will give the value of the product of the base number to itself when multiplied the exponent times. If the exponent of a negative number is given as $(-a)^{b}$ then it implies that (-a) is multiplied by (-a) for b times. Here, the (-) symbol will also be multiplied to (-) symbol $b$ times. Hence, if b is odd then the value of exponent of the negative number will be negative else the value will be positive.In the expression $(-a)^{b}$; (-a) is the base value and b is the power. If the power of an exponent term is zero then its value will be 1. For any number a, $a^{0}=1$.

There can also be a case that the base number is positive and the power is negative. In that case, the negative exponent becomes positive by taking the reciprocal of the expression. For example, $2^{-4}$ = $\frac{1}{2^{4}}$.

## How to Find Exponent of a Negative Number

Suppose, the base number is a < 0 and the power is b then the exponent value of $a^{b}$ can be obtained by following the given steps:

1) Multiply a by a such that E = a*a.

2) Decrease b by 1.

3) E = E*a

4) Repeat steps 2 and 3 till b > 1.

Suppose, we have to find the value of $(-2)^{3}$. Then, we will simply find the value of $-2\times -2\times -2$ but if we need to find the value of $(-2)^{32}$ then it would be better to follow the above algorithmic approach. Let us take an example of $(-3)^{4}$. Here, a = -3, b = 4
Step 1: E = $-3\times -3=9$

Step 2: b = 4-1 = 3

Step 3: E = $9\times -3=-27$

Step 4: b = 3 - 1 = 2

Step 5: E = $-27\times -3=81$

Step 6: b = 2- 1 = 1

As b becomes 1, the final value of E = 81.

## Negative Numbers with Exponents Examples

Example 1: Find the value of $(-7)^{4}$ and give step-wise solution.

Solution:  Let us take, a = -7 and b = 4.

Step 1: E = $-7\times -7=49$

Step 2: b = 4 -1 = 3

Step 3: E = $49\times -7 = -343$

Step 4: b = 3 -1 =2

Step 5: E = $-343\times -7 = 2401$

Step 6: b = 2 - 1 = 1

As b becomes 1, the final value of E = 2401. Hence, $(-7)^{4}=2401$
Example 2: Is the value of $(-77)^{34}\times (-9)^{23} \times 81^{-4}$ positive?

Solution: We have,

$(-77)^{34}$ as positive (+) as the 34 is an even number.

$(-9)^{23}$ as negative (-) as 23 is an odd number.

$81^{-4}$ as positive (+) as 81 is positive.

Hence, the expression $(-77)^{34}\times (-9)^{23} \times 81^{-4}$ will give values like $(+)\times (-)\times (+) = (-)$.

Hence, the value of $(-77)^{34}\times (-9)^{23} \times 81^{-4}$ is not positive.
Example 3: Find the value of
i) $-6^{2}$
ii) $(-5)^{2}$

Solution:

i)
The value of $-6^{2} = -6\times 6$.

Now, as we can see that the power of 2 has been raised to the number 6 but not to the minus sign.

Hence, we can say that $-6^{2} = -36$.

ii) In $(-5)^{2}$ the power of two has been raised to (-5) where the minus sign is included. Hence,
$(-5)^{2} = -5\times -5 = 25$.