# Exponents

**Exponents are short forms used to represent a set of repeated factors.**

$3^{6}$ is used to represent 3 multiplied to itself six times. i.e 3 x 3 x 3 x 3 x 3 x 3. In $3^{6}$, the number 3 is called the Base and the number 6 is called the exponent or power.

The exponent is defined as a number which is used to represent the number of times a given number has to be multiplied by itself. Exponents could also be defined as the operation in raising a particular number to a power in multiplication and in which all the factors are equal.

The expression of 8$^{3}$ means that three 8’s are to be multiplied with each other. Similarly, 6$^{2}$ would mean 6 x 6. The relevant power of any particular number is the number by itself.

The power is defined as the number of times the number is to be taken as one of the factor. The method of finding a root is nothing but the inverse of raising a number to a power.

Exponent is an index of a power. "The exponent is also known as the Powers or Indices".

Consider the following:

2 - two occurs once

2 X 2 - two occurs twice

2 X 2 X 2 - two occurs thrice

We can see that the number 2 is multiplied to itself some number of times. If 2 has to be multiplied to itself many times, it would become tough to write the same. So, there is a need to write in some notation.

So, we write 2 multiplied to itself thrice as $2^{3}$

2 - two occurs once $2^{1}$

2 X 2 - two occurs twice $2^{2}$

2 X 2 X 2 - two occurs thrice $2^{3}$

There are a number of rules of exponents that will help us to solve problems related to exponents with much less effort. The rules of exponents are listed below.

### Rules of exponents:

(a) The product rule for exponents

(b) The quotient rule for exponents

(c) The power rule for exponents

### The product rule of exponents:

To develop a particular rule for multiplying exponential expressions with the same base, we would have to consider the product x$^{2}$ . x$^{4}$ and since the expression x$^{2}$ means that x is to be used as a factor two times and the expressions x$^{4}$ means x is to be used as a factor four times.

x$^{2}$ . x$^{4}$ = (x . x ) (x .x . x. x) = (x .x. x. x. x . x) = x$^{6}$

In general, we have x$^{m}$ . x$^{n}$ = x$^{(m+n)}$

To multiply two exponential expression with the same base, the common bases is kept as it is and the exponents are added.

### The quotient rule for exponents:

When we divide two powers of ‘x’ the number of factors of ‘x’ in the solution are found to be the difference between the factors in the numerator of the fraction and the factors in the denominator.

So whenever we need to divide expressions with the same base we have to keep the base and then subtract the exponent in the denominator from the exponent present in the numerator.

Quotient rule for exponents: $\frac{x^{n}}{x^{m}}$ = x$^{(n - m)}$

**To apply the quotient rule for exponents we need to follow the following steps:**

**Step 1:**Reduce the constants and subtract the exponents using the quotient rule for exponents.

**Step 2:**Take the reciprocal to get rid of the negative exponent, wherever necessary.

**Step 3:**Get the positive exponents in the final answer.

**Example:** Divide x$^{5}$ by x$^{3}$

$\frac{x^{5}}{x^{3}}$ = $\frac{(x . x . x . x . x)}{(x . x . x)}$ = $\frac{(x . x )}{1}$ = x$^{2}$

### The power rule for exponents:

When we have to raise an exponential expression into another power, we need to do the multiplication of the exponents keeping the base the same.

So, $(x^{m})^{n}$ = x$^{mn}$

Let us try one example based on the power rule of exponents:

Find the solution of $(x^{5})^{2}$

**Step 1:** Keeping the base same multiply the exponents.**Step 2:** Write the final exponential power as multiplicative of both the exponents (5 x 2 = 10)

$(x^{5})^{2}$ = x$^{5 x 2}$ = x$^{10}$

Exponentiation is a mathematical operation or process involving two numbers where the components are found to be a base ‘a’ and the exponent ‘n’ with a formal expression of a$^{n}$.

When ‘n’ is found to be a positive integer, the operation exponentiation will correspond to repeated multiplication.

In other words, we could also say it’s a product of ‘n’ factors of ‘a’.

The product may also be referred to as power.

An expression has two parts in exponentiation, a base and a power. The power is often called as the exponent.

The base is a number we could raise to a power. The power is a number we raise the base and it is indicated by a raised number or a superscript.

### Examples of Exponentiation:

Consider 14$^{3}$. It has 14 as the base and 3 as the exponent. We could term it as fourteen raised to the 3rd power.

There are few steps to be followed to add exponents having the same base.

**Step 1:**Take the smallest exponent as common

**Step 2:**Solve it further

There are few steps to be followed to add exponents having a different base.

**Step 1:**Find each exponent value separately

**Step 2:**Add the resultants

### Examples on Adding Exponents:

Given below are some examples on adding exponents.

**Example 1:**

Solve 7^{2} + 7^{4}

**Solution:****Step 1:** Take the smallest exponent as common $\rightarrow$ 7^{2}(1 + 7^{2}) **Step 2:** Solve it further $\rightarrow$ 49(1 + 49) **Step 3:** 49 x 50 = 2450

**Example 2:**

Solve 5^{3} + 2^{4}

**Solution:****Step 1:** Find each exponent value separately $\rightarrow$ (5 x 5 x 5) + (2 x 2 x 2 x 2) = 125 + 16**Step 2:** Add the resultants = 141

There are few steps to be followed to subtract exponents having the same base.

**Step 1:**Take the smallest exponent as common

**Step 2:**Solve it further

There are few steps to be followed to subtract exponents having a different base.

**Step 1:**Find each exponent value separately

**Step 2:**Subtract the resultants

### Examples on Subtracting Exponents:

Given below are some examples on subtracting exponents.

**Example 1:** Solve 7^{3} - 7^{2}**Solution:****Step 1:** Take the smallest exponent as common $\rightarrow$ 7^{2}(7^{1} - 1) **Step 2:** Solve it further $\rightarrow$ 49(7 - 1) **Step 3:** 49 x 6 = 294

**Example 2:** Solve 8^{3} - 5^{2}

**Solution:****Step 1:** Find each exponent value separately $\rightarrow$ (8 x 8 x 8) + (5 x 5) = 512 - 25**Step 2:** Subtract the resultants = 487

Exponents or Indices are used to tell how many times a factor must be multiplied by itself. The factor may be a number (constant) or a variable. Consider $9^{2}$. The factor is 9 and is called as the base and the exponent or the index is 2. It means 9 must be multiplied 2 times 9 × 9.

We can also multiply the factors in an exponential notation. Multiplication of variables or constants with exponents is simple and the process is the same for both numbers and variables. For example,

($2^{2}$)($2^{2}$) = $2 \times 2 \times 2 \times 2$

= $2^{4}$

($x^{2}$)($x^{2}$) = $x \times x \times x \times x$

= $x^{4}$

If m is a positive integer and a in the Real Number Set and a ? 0, then a × a × a × ... m times is $a^{m}$ and a × a × a ×.... n times is $a^{n}$ and the product of $a^{m}$ and $a^{n}$ is

$a^{m}$ × $a^{n}$ = (a × a × a × ... m times) (a × a × a ×.... n times)

$a^{m}$ × $a^{n}$ = $a^{m+n}$

### Multiplication of Indices

To multiply indices, we have to add the exponents!

**Note:** This rule is applicable only when the bases are the same.

We can use this rule for multiplying positive integer exponents, like

$2^{2}$ × $2^{3}$ = $2^{2+3}$= $2^{5}$

We can use the same Product rule to multiply the negative integer exponents

$2^{-2}$ × $2^{3}$ = $2^{-2+3}$ = 2

For any variable x and m, n in the Rational Number set

$x^{m}$ × $x^{n}$ = $x^{m+n}$

The product of $x^{1/2}$ and $x^{3/4}$ is

($x^{1/2}$ )($x^{3/4}$ ) = $x^{(1/2 + 3/4)}$ = $x^{5/2}$

Hence, the rule is applicable to fractional exponents as well! We can also have a combination of constants and variables with exponents.

### Examples on Multiplying Exponents

Given below are some examples that explain the steps to perform multiplication of exponents.

**Example 1:**

Simplify the following algebraic expression with exponents

($x^{3}y^{-2}$)(3x$y^{4}$)

**Solution:**$x^{3}y^{-2}$ = $1x^{3}y^{-2}$

Hence ($x^{3}y^{-2}$)(3x$y^{4}$) = ($1x^{3}y^{-2}$)(3x$y^{4}$)

Multiply the constants and group the x terms and y terms together.

($1x^{3}y^{-2}$)(3x$y^{4}$) = $3(x \times x^{3})(y^{-2} \times y^{4})$

Simplifying using the Product Rule,

$3(x \times x^{3})(y^{-2} \times y^{4})$ = $3x^{1+3} y^{-2+4}$

Simplifying further,

$3x^{1+3} y^{-2+4}$ = $3x^{4}y^{2}$

Hence, $(x^{3}y^{-2})(3xy^{4})$ = $3x^{4}y^{2}$

**Example 2:**

Simplify $(2^{n} \times 8^{2n})16^{3n}$

**Solution:**

We cannot use the product rule of indices to simplify as the bases are not same. But if we observe carefully, it is possible to rewrite each term with base 2.

$8 = 2^{3}$ and $16 = 2^{4}$

$(2^{n} \times 8^{2n})16^{3n}$ = $(2^{n} \times (2^{3^{2n}}))(2^{4^{3n}})$

According to the order of operations, start simplifying the innermost brackets. According to the law of Indices, $a^{m^{n}}$ = $a^{mn}$. Hence $2^{3^{2n}}$ = $2^{3\times2n}$ and $2^{4^{3n}}$ = $2^{4\times3n}$

$(2^{n} \times (2^{3^{2n}}))(2^{4^{3n}})$ = $(2^{n} \times 2^{3\times2n})(2^{4\times3n})$

= $(2^{n} \times 2^{6n})(2^{12n})$

Using the Product Rule, we can simplify further as the bases are the same. The base is 2.

$(2^{n} \times 2^{6n})(2^{12n})$ = $(2^{n+6n})(2^{12n})$

= $(2^{7n})(2^{12n})$

= $2^{7n+12n}$

= $2^{19n}$

Hence, $(2^{n} \times 8^{2n})16^{3n}$ = $2^{19n}$

We can divide integral exponents by expansion and then cancelling. If we have $x^{9}$ ÷ $x^{4}$, then

$y^{9}$ = $y \times y \times y \times y \times y \times y \times y \times y \times y$

$y^{4}$ = $y \times y \times y \times y$

$y^{9}$ ÷ $y^{4}$ = $\frac{y \times y \times y \times y \times y \times y \times y \times y \times y}{y \times y \times y \times y}$

= $y \times y \times y \times y \times y$

There is an easier way to do it. To divide exponents, if the bases are the same, then subtract the powers. It is as simple as that!

$y^{9}$ ÷ $y^{4}$ = $y^{(9-4)}$ = $y^{5}$

The bases are the same. And so, the result also have the same base. The power is the difference of the powers. 9 - 4 = 5

This method works even when exponents are fractions!

But, what if we have $y^{4}$ ÷ $y^{9}$ ?

$y^{4}$ ÷ $y^{9}$ = $\frac{y^4}{y^9}$

Observe that the exponent of the denominator is greater than the exponent of the numerator. In such case the result has numerator as 1.

$\frac{y^4}{y^9}$ = $\frac{1}{y^{9-4}}$ = $\frac{1}{y^5}$

Let us look at another case,

$\frac{y^3}{y^3}$ = $y^{3-3}$ = $y^{0}$ = 1

When the dividend and the divisor have the same exponent and the same base, then the result is 1.

Let us summarize the results. Consider $x^{m}$ and $x^{n}$, where m and n are rational numbers

- If m > n, then $x^{m}$ ÷ $x^{n}$ = $x^{m-n}$
- If m < n, then $x^{m}$ ÷ $x^{n}$ = $\frac{1}{x^{n-m}}$
- If m = n, then $x^{m}$ ÷ $x^{n}$ = 1
- $x^{0}$ = 1
- $1^{m}$ = 1

Remember x ? 0

When dividing the exponents with the same base, subtract the exponents!

### Examples on Dividing Exponents

Given below are some examples that explain the steps to perform the division of exponents.

**Example 1: **

Evaluate $2^{3/4}$ ÷ $2^{1/2}$

**Solution: **

$2^{3/4}$ ÷ $2^{1/2}$ = $2^{3/4}$ / $2^{1/2}$

The bases are the same and $\frac{3}{4}$ > $\frac{1}{2}$. Hence, we have

$2^{3/4}$ ÷ $2^{1/2}$ = $2^{3/4 - 1/2}$

Using the relation, $\frac{x^m}{x^n}$ = $x^{mn}$

$2^{3/4 - 1/2}$ = $2^{1/4}$

Hence $2^{3/4}$ ÷ $2^{1/2}$ = $2^{1/4}$

In fractional exponents, we observe that the exponent is in the fraction form.

Observe the following:

$2^{1/1}$ = 2

$2^{1/2}$ = $\sqrt{2}$ (square root of 2).

### Examples of Fractional Exponents:

1) $a^{1/n}$ [Here, a is called the base and 1/n is called the exponent or power]

= $n\sqrt{a}$ [nth root of a]

2) $3^{1/2}$

= $\sqrt{3}$

The laws of exponents are as follows:

- If there are parentheses, we have to remove them and use law II, law III, and law IV (multiply exponents)
- If there is a need for division, we have to use law V (subtracting exponents)
- If there is a need for multiplication, we have to use law I (adding exponents)
- If there are negative exponents, we have to use the rules of exponents and convert and re-write as positive exponents.

### Tabulation of Exponent Laws:

For any real number 'a' and positive integer 'm', 'a' multiplied 'm' times can be written as,

a x a x a x a x a x a ..... m times = $a^{m}$

All the Laws of Exponents can be derived from the basic laws of multiplication and division.

### Product Rule for Exponents

Let us consider $3^{3}$ x $3^{2}$

$3^{3}$ x $3^{2}$ = $3 \times 3 \times 3 \times 3 \times 3$ = $3^{5}$

This can be rewritten as $3^{3}$ x $3^{2}$ = $3^{(3+2)}$

= $3^{5}$

In general, for any real number 'a' and rational numbers, 'm' and 'n' we have,

$a^{m}$ x $a^{n}$ = (a x a x a x a x .....m times) x (a x a x a x a...... n times)

$a^{m}$ x $a^{n}$ = m factors x n factors

$a^{m}$ x $a^{n}$ = total of (m+n) factors

$a^{m}$ x $a^{n}$ = $a^{(m+n)}$

The product of two exponential numbers with the same base is the base raised to the power of the sum of the exponents. Remember, the bases must be same! We cannot use this rule if the bases are different.

### Quotient Rule for Exponents

Consider $3^{4}$ ÷ $3^{2}$

$3^{4}$ ÷ $3^{2}$ = (3 x 3 x 3 x 3) ÷ (3 x 3)

= 3 x 3

= $3^{2}$

This can be rewritten as $3^{4}$ ÷ $3^{2}$ = $3^{(4-2)}$

= $3^{2}$

So, in order to divide exponents with same base, we have to subtract the exponents. In general, for any real number 'a' and two positive integers 'm' and 'n',

$a^{m}$ ÷ $a^{n}$ = (a x a x a x ...m times) ÷ (a x a x a x .... n times)

$a^{m}$ ÷ $a^{n}$ = a x a x a ..... (m-n) times

$a^{m}$ ÷ $a^{n}$ = $a^{(m-n)}$

This rule sometimes results in a negative exponent, if m < n. If m = n, then the exponent is zero.

For any real number 'a' and two rational numbers 'm' and 'n',

$a^{m}$ ÷ $a^{n}$ = $a^{(m-n)}$ m - n > 0, if m > n

$a^{m}$ ÷ $a^{n}$ = $a^{(m-n)}$ m - n < 0, if m < 0

$a^{m}$ ÷ $a^{n}$ = $a^{0}$ m - n = 0, if m = n

### Power of Exponents

Consider $(3^{4})^{2}$

$(3^{4})^{2}$ = ($3^{4}$) x ($3^{4}$)

= (3 x 3 x 3 x 3) x (3 x 3 x 3 x 3)

= $3^{8}$

This can be rewritten as $(3^{4})^{2}$ = $3^{(4 x 2)}$ = $3^{8}$

In general, for any real number 'a', and rational numbers 'm' and 'n', $(a^{m})^{n}$ = $a^{mn}$

### Power of Product

Consider $(3 \times 4 \times 5)^{3}$ = $(3 \times 4 \times 5) \times (3 \times 4 \times 5) \times (3 \times 4 \times 5)$

= $3^{3} \times 4^{3} \times 5^{3}$

In general, for any real number a, b and c and rational numbers m, $(a x b x c)^{m}$ = $a^{m} \times b^{m} \times c^{m}$

### Exponent of a Quotient

Consider ($\frac{2}{3}$)^{3} = $\frac{2}{3}$ $\times$ $\frac{2}{3}$ $\times$ $\frac{2}{3}$ = $\frac{2^3}{3^3}$

In general, for any two integers a, b and rational numbers m, ($\frac{a}{b}$)^{m} = $\frac{a^m}{b^m}$

Exponent properties cover all the rules in exponentiation that are required to solve a exponential expression.

**Property 1:** If a is any real number and ‘r’ and‘s’ are integers, then a^{r }.a^{s} = a^{r + s} . This is also known as the product of power property.^{ }

**Property 2: **This is also known as a quotient of powers property. If we divide two powers of the same base we have to subtract the exponents.

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

**Property 3: **Power of a product property. If we multiply two powers of same exponent but have different bases then we raise the product to the power.

a^{c} × b^{c} = (ab)^{c}

**Property 4:** Power of power property. If we multiply the whole expression by itself over and over again then it results in raising the expression to a power.

$(a^{c})^{b}$ = a^{cb}

The exponential notation $x^{m}$ indicates that x is multiplied m times. But, when m is negative, we cannot practically multiply a factor negative number of times. It means the opposite of multiplication. The opposite of multiplication is division!

If we have $4^{-3}$, then it means ($\frac{1}{4})^3$ = $\frac{1}{4 \times 4 \times 4}$

In general, $x^{-n}$ = 1/$x^{n}$ = $\frac{1}{x \times x \times x \times .....n terms}$

All the Rules of Exponents are applicable to Negative Exponents as well. For any real number a and positive integers m and n,

a) $a^{m}$ × $a^{n}$ = $a^{m+n}$

$2^{-3}$ × $2^{-4}$ = $2^{-3-4}$ = $2^{-7}$

b) $a^{m}$ ÷ $a^{n}$ = $a^{m-n}$

$2^{-6}$ ÷ $2^{-9}$ = $2^{-6+9}$ = $2^{3}$

c) $a^{m^{n}}$ = $a^{mn}$

$2^{-3^{-4}}$ = $2^{12}$

d) $(ab)^{m}$ = $a^{m}b^{m}$

$(2 \times 3)^{-2}$ = $2^{-2}3^{-2}$

e) $(a/b)^{m}$ = $a^{m}$/$b^{m}$

$(2/3)^{-2}$ = $2^{-2}$/$3^{-2}$

Here is a useful tip:

$a^{-m}$ = 1/$a^{m}$ and also $a^{m}$ = 1/$a^{-m}$

$2^{-3}$ = 1/$2^{3}$ and $2^{3}$ = 1/$2^{-3}$

### Examples on Negative Exponents:

Given below are some examples on negative exponents.

$2^{-3}$ = 1/$2^{3}$

$27^{-13}$ = 1/$27^{13}$

If the exponent is not an integer then we use the product of powers property for solving exponents. Exponents like $\frac{1}{2}$ power works like square root while powers of $\frac{1}{3}$ works out as cube root.

$a^{\frac{1}{2}}$ = $\sqrt{a}$ and similarly we have $a^{\frac{1}{3}}$ = $\sqrt[3]{a}$

### Examples on Rational Exponents:

Given below are examples on rational exponents.

64$^{\frac{1}{3}}$ = $\sqrt[3]{64}$ = 8

9$^{\frac{1}{2}}$ = $\sqrt{9}$ = 3

We could apply the rules for exponents to simplify many expressions involving variable exponents.

### Examples on Variable Exponents:

Given below are some examples that explains how to simplify expressions having variable exponents.

**Example 1:**

Solve $\frac{7^{n}}{7^{n}}$

**Solution:**

**Step 1:** Keeping the base common subtract the exponents**Step 2:** Combine like terms (n - n = 0)

$\frac{7^{n}}{7^{n}}$

7$^{n-n}$

7$^{0}$ = 1

**Example 2: **

Solve (m$^{2x}$) (m$^{3x}$)

**Solution: **

**Step 1:** Keep the common base and then add the exponents**Step 2:** Combine like terms (2x + 3x = 5x)

m$^{(2x+3x)}$ = m$^{5x}$

Simplifying the expressions in exponents with the help of the law of exponents is considered as simplifying exponents.

For positive integer: m$^{n}$ = m. m. m. ….m ( n factors)

Zero: m$^{0}$ = 1

Negative: m$^{-n}$ = $\frac{1}{m^{n}}$ and also, $\frac{1}{m^{-n}}$ = m$^{n}$

Fractional: m$^{(\frac{n}{d})}$ = $\sqrt[d]({m}^{n})$ = $(\sqrt[d]{m})^{n}$

### Examples on Simplifying Exponents:

Given below are some examples on simplifying exponents.

**Example 1: **

Find $\frac{55x^5 y^3}{5x^2 y^5}$

**Solution: **

**Step 1:** Simplify the constants

$\frac{55}{5}$ = 11

**Step 2:**

Simplify the x terms

$\frac{x^5}{x^2}$ = $x^{5-2}$ Since, $\frac{x^m}{x^n}$ = $x^{m-n}$ when m > n

= $x^{3}$

**Step 3:** Simplify the y terms

$\frac{y^3}{y^5}$ = $\frac{1}{y^{5-3}}$ Since $\frac{x^m}{x^n}$ = $\frac{1}{x^{m-n}}$

= $\frac{1}{y^2}$

**Step 4:** Putting them together we have

$\frac{55x^5 y^3}{5x^2 y^5}$ = $\frac{11x^3}{y^2}$

In complex exponents, there are n distinct complex values associated with z$^{\frac{1}{n}}$ , z $\neq$ 0

If we write z = r e$^{i \theta}$ then, z = $r^{\frac{1}{n}}$ $e^{i[\frac{(\theta + 2k \pi)}{n}]}$ is a distinct nth root for k = 0,1,2,3,4…. n-1. The values of $z^{\frac{1}{n}}$ vary as the argument of z taking the values of $\theta$, $\theta$ + 2$\pi$, $\theta$ + 4$\pi$.....$\theta$ + 2(n - 1)$\pi$.

A unique value of $z^{\frac{1}{n}}$ may be of non-zero complex number obtained by restricting the argument and this seems to indicate a link between the nth roots and logarithm.

The function is defined as w = $z^{\frac{1}{n}}$ = $e^{(\frac{1}{n})log z}$ = $e^{\frac{1}{n}(log|z|+ i arg z)}$

This function like logarithm function is found to be multi valued and for any other integer k, one of these n values will be repeated.

Generally if ‘m’ and ‘n’ are positive integers with no common factors we could define this as ($z^{\frac{1}{n}}$) = $e^{(\frac{m}{n})log z}$, z $\neq$ 0.

**Examples of Complex Exponents**

1) 5$^{\frac{1}{2}}$ = e$^{\frac{1}{2} log 5}$ = e$^{\frac{1}{2}(ln 5 + 2k \pi i)}$ = e$^{\frac{1}{2} ln 5}$ e$^{k \pi i}$ = $\pm$ $\sqrt{5}$

2) i$^{\frac{1}{2}}$ = e$^{\frac{1}{2} log i}$ = e$^{\frac{1}{2} i(\frac{\pi}{2} + 2k \pi)}$ = $\pm$ e$^{\frac{\pi i}{4}}$ = $\pm$ $\frac{\sqrt{2}}{2}$ (1 + i)

In an integral exponent the exponent is a whole number which is an integer.

Example: $2^{3}$, $5^{152}$, $100^{-3}$, $21^{-9}$

Consider $5^{1/2}$, $16^{2/3}$. Here, the exponents are not integers. So, these are examples of numbers with non integral exponents.

### Integral Exponent for a positive base:

For a positive integer a and n, we have,

$a^{n}$ is always positive

and

$a^{-n}$ = $\frac{1}{a^{n}}$ , which is always positive

### Positive Integral Exponent for a negative base:

For a positive integer a and n, we have

$(-a)^{n}$ [Here (-a) is the base and n is the exponent or power]

= {$a^{n}$ , when n is even}

= {$(-a)^{n}$ , when n is odd}

**For example:**

(-2) x (-2) ..... Here, -2 occurs twice. So, = $(-2)^{2}$ = 4

(**Note:** Two negative becomes positive)

(-2) x (-2) x (-2)..... Here, -2 occurs thrice. So, $(-2)^{3}$ = -8

(**Note:** Two negatives becomes positive and one negative remains. So, it will be -8)

(-2) x (-2) x (-2) x (-2)..... Here, -2 occurs four times. So, $(-2)^{4}$ = 16

(**Note: **Four negatives becomes 2 positives).

### Negative Integral Exponent for a negative base:

For a negative integer a and n, we have

$(-a)^{-n}$ [Here (-a) is the base and (-n) is the exponent]

{$a^{-n}$ = $\frac{1}{a^{n}}$ , when n is even}

{$(-a)^{-n}$ = $\frac{1}{(-a)^{n}}$ , when n is odd}

**For example:**

$(-5)^{-3}$ = $\frac{1}{-5} \times \frac{1}{-5} \times \frac{1}{-5}$

= $\frac{1}{-125}$

= -0.008