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The exponent of a number shows the number of times a number has been multiplied by itself. Exponents help us to write a large multiplication equation involving writing the same number multiple times in a shorter format. Exponents can be positive, negative or a fraction. In this article we will learn about the different formulas that we use in exponents....Read MoreRead Less

An exponent represents the number of times a number is multiplied by itself.

For example, if 7 is multiplied by itself for n times, this can be represented as:

7 x 7 x 7 x 7 x 7….. x n times = 7\(^n\)

In the above expression, 7\(^n\), is written as 7 multiplied by itself n times. Here 7 is called base and n is called exponent.

The following properties are used to solve exponents.

**Product of Power Property**

When we multiply the power with the same base, we add the exponents of the base.

a\(^m\) . a\(^n\) = a\(^{m~+~n}\)

**Power of a Power Property**

When we have to find out the result of the power of a power, we multiply the exponents of the base.

\(\left(a^m\right)^n\) = a\(^{mn}\)

**Power of a Product Property**

When we have to find out the result of the power of a product of two or more numbers, we find the power of numbers and multiply them.

(a . b)\(^n\) = a\(^n\) . b\(^n\)

**Quotient of Power Property**

When we divide the power with the same base, we subtract the exponents of the base.

\(\frac{a^m}{a^n}\) = \(a^{a+b}\)

**Zero Exponent**

When we find that the power is zero of any non zero number, we get the result as 1.

\(a^0\) = 1

**Negative Exponent**

When we have a negative exponent of any non zero number, we get the result as the reciprocal of power with a positive exponent.

a\(^{-n}\) = \(\frac{1}{a^n}\)

**Formulas of Exponents**

- a\(^m\) . a\(^n\) = a\(^{m~+~n}\)

- \(\left(a^m\right)^n\) = a\(^{mn}\)

- (a . b)\(^n\) = a\(^n\) . b\(^n\)

- \(\frac{a^m}{a^n}\) = \(a^{a+b}\)

- \(a^0\) = 1

- a\(^{-n}\) = \(\frac{1}{a^n}\)

Where a is base of power, and m and n are the exponents of power.

**Example 1:** Simplify the expression: \(\frac{4^5~.~3^3}{12^3}\).

**Solution:**

\(\frac{4^5~.~3^3}{12^3}\) [Write the expression]

= \(\frac{4^5~.~3^3}{(4~.~3)^3}\) [Power of Product property ]

= \(\frac{4^5~.~3^3}{4^3~.~3^3}\) [Simplify]

= 4\(^{5-3}\) . 3\(^{3-3}\) [Quotient of Power property ]

= 4\(^2\) . 3\(^0\) [Simplify]

= 4\(^2\) . 1 [Zero exponents]

= 16 [Simplify]

**Example 2:** Simplifying the expression: \(\frac{7^4~.~(x~.~7)^7~.~x^2}{7^{-3}~.~x^4}\).

**Solution:**

\(\frac{7^4~.~(x~.~7)^7~.~x^2}{7^{-3}~.~x^4}\) = \(\frac{7^4~.~x^7~.~7^7~.~x^2}{7^{-3}~.~x^4}\) [Power of Product property]

= \(\frac{7^{4+7}~.~x^{7+2}}{7^{-3}~.~x^4}\) [Product of Power property]

= \(\frac{7^{11}~.~x^{9}}{7^{-3}~.~x^4}\) [Simplify]

= \(\frac{7^{11}~.~x^{9-4}}{7^{-3}}\) [Quotient of Power property ]

= \(\frac{7^{11}~.~x^{5}}{7^{-3}}\) [Simplify]

= \(7^{11}~.~x^5~.~7^3\) [Negative exponents]

= \(7^{11~+~3}~.~x^5\) [Product of Power property]

= \(7^{14}~.~x^5\) [Simplify]

**Example 3: **In a sculptor’s workshop there are multiple sculptures of the same type. The smallest sculpture is 2 cm tall. The height of each successive sculpture in the workshop is twice the height of the previous sculpture. What are the different heights of the figures in the workshop if the height of the tallest sculpture is 256 cm?

**Solution:**

According to the question and the image, the smallest sculpture is 2 cm and the largest sculpture has a height of 256 cm. Since the height is progressing by multiplying the previous height by two, these are the heights of the sculptures:

1st sculpture: 2 cm

2nd sculpture: 2\(^2\) = 2 x 2 = 4 cm

3rd sculpture: 2\(^3\) = 2 x 2 x 2 = 8 cm

4th sculpture: 2\(^4\) = 2 x 2 x 2 x 2 = 16 cm

5th sculpture: 2\(^5\) = 2 x 2 x 2 x 2 x 2 = 32 cm

6th sculpture: 2\(^6\) = 2 x 2 x 2 x 2 x 2 x 2 = 64 cm

7th sculpture: 2\(^7\) = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 cm

8th sculpture: 2\(^8\) = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 cm

So, the different heights of the sculptures in the workshop are 2 cm, 4 cm, 8 cm, 16 cm, 32 cm, 64 cm, 128 cm and 256 cm. In terms of the powers of 2, the sculptures are ordered as, 2\(^1\) cm, 2\(^2\) cm, 2\(^3\) cm, 2\(^4\) cm, 2\(^5\) cm, 2\(^6\) cm, 2\(^7\) cm, 2\(^8\) cm.

Frequently Asked Questions

An exponent represents the number of times a number is multiplied by itself.

When we have to find out the result of the power of a product of two or more numbers, we find the power of each number and multiply them.

When we have to find out the result of the power of a power, then we multiply the exponents of the base.

A number when raised to 1 denotes the number itself.