Evaluating Powers and Exponents (Definition, Types and Examples) - BYJUS

# Evaluating Powers and Exponents

Power is an alternate method of expressing repeating multiplication of the same number in an expression. The exponent represents the number of times the number is multiplied by itself. Learn how to evaluate the value of powers with the help of some solved examples....Read MoreRead Less ## Evaluating Powers and Exponents

Exponents and powers are used to make the writing of long, repeated multiplication processes simpler. If we need to express 5 × 5 × 5 × 5 in an easier  way, we can write it as $$5^4$$, where 4 is the exponent and 5 is the base. Power is the value of $$5^4$$

In its most basic form, power is an expression that shows the multiplication of the same number or factor over and over again. The exponent is the number of times the base is multiplied by itself. Here are some examples:

$$3^3$$ is 3 raised to the power 3 therefore, $$3^3$$ =  3 × 3 × 3 = 27

$$2^2$$ is 2 raised to the power 2 therefore, $$2^2$$ = 2 × 2 = 4.

## What is Power?

Power is a product of repeated factors. The repeated factor is the power’s base. ## What is an Exponent?

An exponent represents the number of times a number is multiplied by itself. When 5 is multiplied by itself n times, it is written as $$5\times5\times5\times5\times5\times\ldots~n~ times=5^n$$

The above expression, $$5^n$$, is written as 5 multiplied by itself n times. As a result, exponents are also known as power or indices.

For example:

• $$3\times3\times3\times3=3^4$$
• $$4\times4\times4=4^3$$
• $$10\times10\times10\times10\times10=10^5$$

## What is the Dot Symbol?

The (⋅) symbol is used between two mathematical expressions to indicate that the second expression is being multiplied by the first. It is similar to the “×” symbol we use for multiplication.

For example, $$4\cdot 4\cdot 4=4^3$$

## Perfect Square

A perfect square is a number that can be expressed as the product of an integer by itself or as the integer’s second exponent. For example, 64 is a perfect square because it is the product of the integers 8 and 8, that is, $$8\times8=8^2$$ which equals 64.

Examine each number to see if it is a perfect square.

1. 81

Because $$9^2=81,81$$ is a perfect square.

2. 10

No whole number squared equals 10.

So, 10 is not a perfect square.

## How is the Repeated Multiplication of the Same Number Different from the Repeated Addition of the Same Number?

Repeated addition is the process of combining equal groups of numbers. Multiplication is another term for it. When the same number is repeated, we can write it as multiplication.

For instance, 2 + 2 + 2 + 2 + 2. Since 2 is repeated 5 times in this addition, we can write it as $$5\times2=10$$.

However, the repeated multiplication of 2 is $$2\times2\times2\times2\times2=2^5=32$$ where 2 is the base and 5 is the exponent. The exponent 5 indicates that 2 has been multiplied by itself five times.

## How to use a Scientific Calculator to find the Power

• In the scientific calculator, enter a number that is the base of the power.

• Press the button that has the symbol “^” on it.

• Now, enter the value of the exponent.

• The answer will be displayed.

For example: Press 4 followed by ^ and 3 to get the result of 4 to the third power. 64 should be displayed as the answer.

Another way to find power:

• In the scientific calculator, enter the number that is the base of the power.

• In the calculator, press the button that has the  symbol $$x ^{[]}$$ on it.

• Now, enter the value of the exponent.

• Post that, press the button that has the symbol “=” on it.

• The answer will come into view.

For example: Press 2 followed by $$x ^{[]}$$ and then press 3 to get the result of 2 to the third power. The answer should be 8.

## Solved Examples

Example 1:

A life-size  board game is a square with a side length of 12 yards. What is the area of the board game? Solution:

Given that, the side length of the square board game is 12 yards.

$$\text{Area of board game} = \text{(side length)}^2$$

$$=(12)^2$$

$$=12\times12$$

$$=144$$

Hence, the area of the board game is 144 square yards.

Example 2:

A square painting measures 2 meters on each side. What is the area of the painting in square centimeters?

Solution: Given that, the side length of the painting is 2 meters.

$$\text{Area of painting} = \text{(side length)}^2$$

$$= (2)^2$$

$$=2\times2$$

$$= 4$$ square meters

As we know, 1 meter = 100 centimeters.

$$= 4\times 100^2$$ centimeters.

$$= 4\times100\times100$$ square centimeters.

$$= 40,000$$ square centimeters.

Hence, the area of the painting is $$40,000$$ square centimeters.

The term “power” refers to the repeated multiplication of a value or integer. For example, $$a^n$$ is a power in which the base is a and the exponent is n.

The base of a power is the repeated factor, whereas the exponent of the power is the number of times the base is used as a factor.

For example, $$7^3$$ is the power which shows that 7 is multiplied three times and 3 is the exponent.

We get 1 as the final result if the exponent is 0. For example, in the case, $$8^0=1$$

When the exponent is one, the value of the base remains unchanged. For instance, $$8^1=8$$ .

The following are some examples of exponents:

$$3\times3\times3=3^3$$ (3 raised to the 3rd power)

$$7\times7\times7\times7=7^4$$ (7 raised to the 4th power)

$$6\times6\times6\times6\times6=6^5$$ (6 raised to the 5th power)