# How to calculate Exponent values? With examples

How to calculate Exponent values An exponent is a tool used to answer a variety of algebraic problems. It is used to calculate quantities that are either huge or very little. By grabbing the knowledge of basic exponents rules will make your math studies more effective and enjoyable.

In this article, we will discuss the definition of exponents and how to calculate the problems involving exponents?

## What is the exponent?

The exponent of a number shows how many times the number is multiplied by itself. The number which is written in power is said to be the base and power is called index, for example, 5 x 5 x 5 can also be written as 5^{3} because 5 is multiplied by itself three times as it holds the definition of the exponent. In this example, 5 is the base, and 3 is the power or exponent or index. In general, a^{m} means that a will be multiplied by itself for **m** times.

a^{m} = a x a x a x … x a (n times)

## How to calculate Exponents?

To calculate an exponents-related problem, you must learn some basic rules that are necessary to solve a problem accurately. Some rules are given below.

- Whenever two numbers with the same base having exponents can be multiplied then bases remain the same and powers will be added. For example, 2
^{3}x 2^{4}= 2^{3+4}= 2^{7}.

In general, a^{n} x a^{m} = a^{n+m}.

- Whenever two numbers with the same base having exponents can be divided then bases remain the same and powers will be subtracted. For example, 2
^{5}/2^{3}= 2^{5-3}= 2^{2}. In general, a^{n}/ a^{m}= a^{n-m}. - When an exponent number has power, simply multiply the powers. For example, (5
^{4})^{2}= 5^{4×2}= 5^{8}. In general, (a^{n})^{m}= a^{nxm}. - When a product of two numbers has an exponent, then that exponent will be applied to both the numbers separately. For example, (4 x 5)
^{2}= 4^{2}x 5^{2}. In general, (ab)^{n}= a^{n}x b^{n}. - When a quotient of two numbers has an exponent, then that exponent will be applied to both the numbers separately. For example, (4 / 5)
^{2}= 4^{2}/ 5^{2}. In general, (a/b)^{n}= a^{n}/ b^{n}. - When a base is raised to a power of zero it is always equal to 1. For example, 2
^{0}= 1.

In general, a^{0} = 1

- When an exponent has negative power, reciprocal the number to make it positive. For example, 3
^{-4}= 1/3^{4}In general, a^{-n}= 1/a^{n}.

Let us take some examples to understand this concept.

**Example 1**

Evaluate **5 ^{3}**.

**Solution**

This is a simple exponent problem just multiply the base for exponents times.

**Step 1: **write the base multiply by itself in exponent times.

**5 ^{3}**= 5 x 5 x 5

**Step 2: **Multiply to get the result.

**5 ^{3}**= 5 x 5 x 5 = 125

**Example 2**

Evaluate **6 ^{3} x 6^{2}.**

**Solution**

This is the product of two exponent values, in this type of problem bases, remain the same and powers should be added.

**Step 1: **Apply the product rule.

6^{3} x 6^{2} = 6^{3+2}

= 6^{5}

**Step 2: **write the base multiply by itself in exponent times.

**6 ^{5} = **6 x 6 x 6 x 6 x 6

**Step 3: **Multiply to get the result.

**6 ^{5} = **6 x 6 x 6 x 6 x 6 = 7776

You can verify the result by using Exponent Calculator.

**Example 3**

Evaluate **2 ^{5} / 2^{2}.**

**Solution**

This is the quotient of two exponent values, in this type of problem bases remain the same and powers should be subtracted.

**Step 1: **Apply the quotient rule.

2^{5 }/ 2^{2} = 2^{5-2}

= 2^{3}

**Step 2: **write the base multiply by itself in exponent times.

**2 ^{3} = **2 x 2 x 2

**Step 3: **Multiply to get the result.

**2 ^{3} = **2 x 2 x 2 = 8

**Example 4**

Evaluate (**4 ^{3})^{2}.**

**Solution**

**Step 1: **Apply the power of power rule.

(**4 ^{3})^{2} **= 4

^{3×2}

= 4^{5}

**Step 2: **write the base multiply by itself in exponent times.

**4 ^{5} = **4 x 4 x 4 x 4 x 4

**Step 3: **Multiply to get the result.

**4 ^{5} = **4 x 4 x 4 x 4 x 4 = 1024

**Example 5**

Evaluate (**3 x 4) ^{2}.**

**Solution**

**Step 1: **Apply the power of product rule.

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

^{2}x 4

^{2}

**Step 2: **write the base multiply by itself in exponent times.

**3 ^{2} = **3 x 3

**4 ^{2} = **4 x 4

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

**Step 3: **Multiply to get the result.

(**3 x 4) ^{2}** = 3 x 3 x 4 x 4 = 9 x 16 = 144

**Example 6**

Evaluate (**2 / 4) ^{2}.**

**Solution**

**Step 1: **Apply the power of quotient rule.

(2 / 4)^{2} **= **2^{2 }/ 4^{2}

**Step 2: **write the base multiply by itself in exponent times

**2 ^{2} = **2 x 2

**4 ^{2} = **4 x 4

(**2 / 4) ^{2}** = 2 x 2 / 4 x 4

**Step 3: **Multiply

(**3 / 4) ^{2}** = 2 x 2 / 4 x 4 = 4 / 16

**Step 4:** Now divide.

(**3 / 4) ^{2}** = 4 / 16 = 1/4 = 0.25

**Example 7**

Evaluate (**7) ^{-3}.**

**Solution**

**Step 1: **Apply the negative power rule.

(**7) ^{-3} **= 1/7

^{3}

**Step 2: **write the base multiply by itself in exponent times.

**1/7 ^{3} = **1/ (7 x 7 x 7)

**Step 3: **Multiply

**1/7 ^{3} = **1/ (7 x 7 x 7) = 1 / 343

**Step 4:** Now divide.

**1/7 ^{3} = **1 / 343 = 0.003

## Summary

Exponents are used widely in scientific notation, standard form, and many other branches of mathematics. Exponents can be calculated easily. Exponents have several rules that are necessary for the calculation of the exponents. These rules are product rule, quotient rule, power of product rule, power of quotient rule, power of a power rule, zero rule, and negative exponent rules. Once you grab the basic concept along with the rules of exponents you can easily solve any problem related to exponents.