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## Exponents Formula

The exponent of a number refers to the number of times any number is multiplied by itself. There are various exponents formulas used to solve the equations. Exponents are important because they help in representing products where a number is repeated by itself many times. Let’s learn about exponents formulas with a few solved examples in the end.

## What Are the Exponents Formulas?

Exponents formulas refer to the formulas that help solve exponents. Exponent of a number is represented in the form: x^{n}, meaning x is multiplied by itself for n times.^{ }Here,

- x is called the “base”
- n is called the “exponent” or “power”
- x
^{n}is read as “x to the power of n” (or) “x raised to n”

### Exponents Formulas

Exponents formulas are expressed as:

- a
^{0 }= 1 - a
^{1}= a - a
^{m }× a^{n }= a^{m+n} - a
^{m }/ a^{n }= a^{m−n} - a
^{−m }= 1/a^{m } - (a
^{m})^{n }= a^{mn} - (ab)
^{m }= a^{m}b^{m} - (a/b)
^{m }= a^{m}/b^{m}

Let us understand exponents formulas better using a few solved examples.

## Examples Using Exponents Formulas

**Example 1: **In a forest, each tree has about 5^{7} leaves and there are about 5^{3} trees in the forest. Using the exponents’ formula, find the total number of leaves.

**Solution: **

To find: Total number of leaves.

The number of trees in the forest = 5^{3}

The number of leaves in each tree = 5^{7}(given)

Using the exponents formula,

a^{x} x a^{y} = a^{x+y}

Putting the values,

5^{3} x 5^{7} = 5^{3+7}

5^{3} x 5^{7} = 5^{10}

**Answer: The total number of leaves is 5 ^{10}.**

**Example 2:** The dimensions of a wardrobe are x^{5} in, y^{3} in, and x^{8} in. Find its volume.

**Solution: **

To find: volume of the wardrobe.

The dimensions of the wardrobe are: length(l) = x^{5} in, width(w) = y^{3} in, height(h) = x^{8} in (given)

Using the exponents formula,

a^{x} x a^{y} = a^{x+y}

Putting the values,

Volume = x^{5} × x^{8 }× y^{3} = x^{13 }× y^{3}

Volume = x^{13 }× y^{3}

**Answer: The volume of the wardrobe is x ^{13} × y^{3}.**

**Example 3: **Determine the value of x^{-5/2} when x = 3.

**Solution: **

To find: the value of x^{-5/2}

Given: x = 3

Using the exponents formula,

x^{-5/2 }= (3)^{-5/2}

= (1/3)^{5/2}

= (1/3 × 1/3 × 1/3 × 1/3 × 1/3)^{1/2}

= (1/243)^{1/2}

= √(1/243) = 1/9√3.

**Answer: The value of x ^{-5/2} when x = 3 is 1/9√3.**

## FAQs on Exponents Formulas

### What Are Exponents Formulas in Math?

We know that the exponent of a number is represented in the form: x^{n} (n is the exponent). In other words, you can say that exponents are superscript numerals. Exponents formulas are the formulas that help solve problems that involve exponents. Some important exponents formulas are given as,

- a
^{0 }= 1 - a
^{1}= a - a
^{m }× a^{n }= a^{m+n} - a
^{m }/ a^{n }= a^{m−n} - a
^{−m }= 1/a^{m } - (a
^{m})^{n }= a^{mn} - (ab)
^{m }= a^{m}b^{m} - (a/b)
^{m }= a^{m}/b^{m}

### What Are the Applications of Exponents Formulas?

Exponents formulas have a wide range of applications:

- scientific scales like the pH scale or the Richter scale.
- calculating the area, volume, and other such problems.
- abbreviates something that would otherwise be really tedious to write.
- used in computer games, measuring scales.
- Science, Engineering, Economics, Accounting, Finance.
- Exponents are often used to describe a computer’s memory.
- Making medicines in the laboratory.

### How To Use Exponents Formulas?

Exponent formulas are easy to use

- Step 1: Check for the given parameters.
- Step 2: Identify the suitable exponent formula.
- Step 3: Put the given values(base and power) in the formula.

### What Are the Components of the Exponent Formulas?

The exponent formulas include the bases and the powers and the mathematical symbols. Exponent of a number is represented as, x^{n}, where ‘x’ is base and ‘n’ is the power.

## Exponents and Powers Formulas

If you time and again wonder if what is an exponent in math, then know an Exponent, also known as power, is a mathematical mannerism of expressing a number multiplied by itself by a certain definite of times. In mathematical terms, when we write a non integer number a, it is actually a^{1}, referred as a — to the power 1.

a^{2} = a*a

a^{3} = a*a*a

a^{4} = a*a*a*a

a^{5} = a*a*a*a

:

:

a^{n} = a*a*a*a*a*a*a . . . n times.

The following is the standard formula of exponent and power to solve problems on exponents.

(am)n = (an)m = amn

The given chart splits the parts in an exponential expression, defining exactly which number is the exponential power to the factor.

In reading problems, mathematical expressions with exponential powers such as7^{4} are often pronounced “seven to the fourth power.” Then again, exponential expressions such as 7^{4} are often read as “the 4^{th} power of 7″.

**Laws of Exponents Formulas**

There are various laws of exponents that you should practice and remember in order to thoroughly understand the exponential concepts. The following exponent law is detailed with examples on exponential powers and radicals and roots.

**Solving Exponential Equations Using Laws**

As now you are already aware that possess one or more terms with a base that is raised to a power ≠ 1. While there is no definite formula for solving an exponential equation, the following rule will help you clearly understand different ways of finding the unknown value in an exponential equation.

**Multiplication of Exponent Rule**

To solve exponential equations, the following are the most important formulas that can be used to multiply the exponents together.

x^{n} x^{m} = x ^{n+m}

x^{n} y^{n = }(xy)^{n}

Now taking the following example with the above power and exponent formula:

4x^{2 }2x^{3 }8x^{-4} = (2^{3})^{2}

4×2×8× {x^{2}x^{3}x^{-4}} = 2^{6}

4×2×8× x^{2+3-4} = 64

8 ×8 ×x^{1} = 64

64 x = 64

Thus, x = 1

**Multiplication of Exponent Rule**

The given formulas are applicable in dividing exponents, particularly these two formulas.

{x/ y}^{n }=x^{n }/ y^{n}

x^{-n }= 1/ x^{n}

Now taking the following example with the above exponent formula will help understanding how to divide exponents.

X^{4}/ y^{3}/ y^{5}/ x^{2 }= ¼

**Isolation and Raise to the Inverse Exponent Rule**

We have to organize the term with an exponent on one side while the other terms on the other side of the equation. Next is to raise both sides of the equation to the inverse exponent.

4x^{4} + 8 = 72

Isolate the term x^{4} by subtracting 8 from both sides and then divide both sides by 4.

4x^{4} = 72 – 8

x^{4} = 16

Now, to isolate x, since x is raised to 4/1^{th} power, raise both sides of the exponential equation to the inverse power (1/4).

x^{4} = 16

{x^{4}}^{(¼)}= 16^{(¼)}

Hence, we get x^{4} = 2

**Factorization**

Solving an equation by isolating an exponent makes it easier to deal with exponent problems. You have to arrange all identical terms on one side of the (=) sign and then factor it.

2x^{2} = 2x + 12 = 16 ……. 1

At this step, divide each term by common factor (i.e. 2) and then subtract the number on the right side.

x^{2} – x + 6 = 8 ………2

x^{2} – x – 2 = 0 ……….3

Apply rule of factoring, simplify and solve

We obtain

(x-2) (x+1)

X= +2, -1.

**Solved Examples of Exponents and Powers Formulas**

**Example1:**

## FAQs (Frequently Asked Questions)

**1. What is Power and Exponent?**

Ans.

**Base –**the number which is multiplied by itself a certain number of times and is used as a factor is called a base. For example: in the expression 5^{4}, the number 5 is called the base, and the number 4 is called the exponent or power. The exponent tally perfectly to the number of times the base is used as a factor.**Exponent –**We exactly know how to calculate the expression 3 x 3. This expression can also be written in a shorter way using something called exponents. 3.3 = 3^{2}. Here in the expression 2 is the exponent. Thus, the number of times a value is multiplied by itself is what we call an exponent.**Power –**It is actually a synonym for exponent, which describes an exponential expression that’s simply multiplication repeated. For example, if we are to say, “Raise 4 to the 7^{th}power,” the base here would be 4 and the power (or exponent to which 4 is raised) would be 7.

2. What is Meant by Radical and Root in an Exponential Equation?

Ans. Radical is a mathematical symbol that is used to denote the square or nth root of a number. As an example, the value of “radical 16” is 4 and the value of “radical 64” is 8.

## Exponents and Powers Formulas

Can we read 10,000,000,000,000,000? These massive natural numbers are not easy to read as well as to recognize and hence evaluate. Exponents make it easy to read and handle very large numbers. We also call exponents, powers or indices in mathematics. Exponents and powers make the complex computations easy and faster. In this topic, we will discuss various exponents and powers formulas with examples. Also, the laws of exponents will be discussed. Let us learn it!

**Exponents and Powers Formulas**

**What are Exponents and Powers?**

We know how to calculate the expression like 6 × 6. This expression can be written in a shorter way using the exponents.So, 6×6=6^{2}. Also, 5×4=5^{4}

An expression that represents repeated multiplication of the same factor is power. The number 6 is the base, and the number 2 is its exponent. The exponent corresponds to the number of times the base will be multiplied by itself.

Therefore, if two powers have the same base then we can multiply these two powers. When we multiply two powers, we will add their exponents. If two powers have the same base then we can divide the powers also. When we divide these powers then we subtract their exponents. Here in this topic, the student will learn about various formulas of exponents.

When we need to repeatedly multiply some number by itself, then it means that we raise it some power. We know this as the Exponent. The power in the exponent represents the number of times that we want to carry out the multiplication operation. Exponents are having their own set of rules when it comes to carrying out Arithmetic Operations. These formulas will help a lot with many mathematical simplifications.

**The Exponents and Powers Formulas are**

Laws of exponents are used to simplify the calculations. Some important laws of exponents are:

**Solved Examples for Exponents and Powers Formulas**

Example-1: Solve 4^{−3}

Solution: As per the the negative exponent rule:

### Basic rules for exponentiation

If n is a positive integer and x is any real number, then x^{n} corresponds to repeated multiplication

To see this rule, we just expand out what the exponents mean. Let’s start out with a couple simple examples.

The general case works the same way. We just need to keep track of the number of factors we have.

##### Quotient of exponentials with same base

If we take the quotient of two exponentials with the same base, we simply subtract the exponents:

This rule results from canceling common factors in the numerator and denominator. For example:

To show this in general, we look at two different cases. If we imagine that a>b, then this rule follows from canceling the common b factors of x that occur in both the numerator and denominator. We are left with just b−a factors of x in the numerator.

If a<b, then what happens? We cancel all the x’s from the numerator and are left with b−a of them in the denominator.

To make the above rule work for this case, we must define a negative exponent to mean a power in the denominator. If nn is a positive integer, we define

Then the rule for the quotient of exponentials works even if a<b:

When b>a, the exponent a−b is a negative number. Since formula (2) is the same no matter the relationship between aa and bb, we don’t need to worry about it and can just subtract the exponents.

#### Power of a power

We can raise exponential to another power, or take a power of a power. The result is a single exponential where the power is the product of the original exponents:

In the last step, we had to remember that multiplication can be defined as repeated addition.

##### Power of a product

If we take the power of a product, we can distribute the exponent over the different factors:

We can show this rule in the same way as we show that you can distribute multiplication over addition. One way to show this distributive law for multiplication is is to remember that multiplication is defined as repeated addition:

but this is NOT equal to

32+52=9+25=34.

#### Special cases

The following are special cases that follow from the rules.

##### The power of one

The simplest special case is that raising any number to the power of 1 doesn’t do anything:

x^{1}=x.

##### The power of zero

As long as x isn’t zero, raising it to the power of zero must be 1:

x^{0}=1.

We can see this, for example, from the quotient rule, as

The expression 0^{0} is indeterminate. You can see that it must be indeterminate, because you can come up with good reasons for it to be two different values.

#### Fractional exponents

The power of power rule (4) allows us to define fractional exponents. For example, rule (4) tells us that

## Exponent Rules: 7 Key Strategies to Solve Tough Equations

The **exponent rules** explain how to solve various equations that — as you might expect — have exponents in them. But there are several different kinds of exponent equations and exponential expressions, which can seem daunting… at first.

Mastering these basic exponent rules along with basic rules of logarithms (also known as “log rules”) will make your study of algebra very productive and enjoyable. Keep in mind that during this process, the order of operations will still apply.

## What are exponents?

Exponents, also known as powers, are values that show how many times to multiply a base number by itself. For example, 4^{3} is telling you to multiply **four** by itself **three** times.

4^{3 }= 4 × 4 × 4 = 64

The number being raised by a power is known as the **base**, while the superscript number above it is the **exponent** or **power**.

The equation above is said as “four to the power of three”. The power of two can also be said as “**squared**” and the power of three can be said as “**cubed**”. These terms are often used when finding the area or volume of various shapes.

Writing a number in exponential form refers to simplifying it to a base with a power. For example, turning **5 × 5 × 5** into exponential form looks like **5 ^{3}**.

Exponents are a way to simplify equations to make them easier to read. This becomes especially important when you’re dealing with variables such as ‘𝒙’ and ‘𝑦’ — as **𝒙 ^{7} × 𝑦^{5} = ?** is easier to read than

**(𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝑦)(𝑦)(𝑦)(𝑦)(𝑦) = ?**

## Rules of exponents in everyday life

Not only will understanding exponent properties help you to solve various algebraic problems, exponents are also used in a practical manner in everyday life when calculating square feet, square meters, and even cubic centimeters.

Exponent rules also simplify calculating extremely large or extremely tiny quantities.These are also used in the world of computers and technology when describing megabytes, gigabytes, and terabytes.

## What are the different rules of exponents?

There are **seven** exponent rules, or laws of exponents, that your students need to learn. Each rule shows how to solve different types of math equations and how to add, subtract, multiply and divide exponents.

Make sure you go over each exponent rule thoroughly in class, as each one plays an important role in solving exponent based equations.

### 1. Product of powers rule

When multiplying two bases of the same value, keep the bases the same and then add the exponents together to get the solution.

4^{2} × 4^{5} = ?

Since the base values are both four, keep them the same and then add the exponents (2 + 5) together.4^{2}× 4^{5} = 4^{7}

Then multiply four by itself seven times to get the answer.

4^{7} = 4 × 4 × 4 × 4 × 4 × 4 × 4 = 16,384

Let’s expand the above equation to see how this rule works:

In an equation like this, adding the exponents together is a shortcut to get the answer.

Here’s a more complicated question to try:

(4𝒙^{2})(2𝒙^{3}) = ?

Multiply the coefficients together (four and two), as they are not the same base. Then keep the ‘𝒙’ the same and add the exponents.

(4𝒙^{2})(2𝒙^{3}) = 8𝒙^{5}

### 2. Quotient of powers rule

Multiplication and division are opposites of each other — much the same, the quotient rule acts as the opposite of the product rule.

When dividing two bases of the same value, keep the base the same, and then subtract the exponent values.

5^{5} ÷ 5^{3} = ?

Both bases in this equation are five, which means they stay the same. Then, take the exponents and subtract the divisor from the dividend.

5^{5} ÷ 5^{3} = 5^{2}

Finally, simplify the equation if needed:

5^{2} = 5 × 5 = 25

Once again, expanding the equation shows us that this shortcut gives the correct answer:

Take a look at this more complicated example:

5𝒙^{4}/10𝒙^{2} = ?

The like variables in the denominator cancel out those in the numerator. You can show your students this by crossing out an equal number of 𝒙’s from the top and bottom of the fraction.

5𝒙^{4}/10𝒙^{2} = 5𝒙~~𝒙~~/10~~𝒙~~

Then simplify where possible, as you would with any fraction. Five can go into ten, five times turning the fraction into ½ with the remaining 𝒙 variables.

5𝒙^{4}/10𝒙^{2} = 1𝒙^{2}/2 = 𝒙^{2}/2

### 3. Power of a power rule

This rule shows how to solve equations where a power is being **raised** by another power.

(𝒙^{3})^{3} = ?

In equations like the one above, multiply the exponents together and keep the base the same.

(𝒙^{3})^{3} = 𝒙^{9}

Take a look at the expanded equation to see how this works:

### 4. Power of a product rule

When any base is being multiplied by an exponent, **distribute** the exponent to **each part** of the base.

(𝒙𝑦)^{3} = ?

In this equation, the power of three needs to be distributed to both the 𝒙 and the 𝑦 variables.

(𝒙𝑦)^{3} = 𝒙^{3}𝑦^{3}

This rule applies if there are exponents attached to the base as well.

(𝒙^{2}𝑦^{2})^{3} = 𝒙^{6}𝑦^{6}

Expanded, the equation would look like this:

Both of the variables are **squared** in this equation and are being **raised** to the power of three. That means three is multiplied to the exponents in both variables turning them into variables that are raised to the power of six.

### 5. Power of a quotient rule

A quotient simply means that you’re dividing two quantities. In this rule, you’re **raising a quotient** by a power. Like the power of a product rule, the exponent needs to be distributed to all values within the brackets it’s attached to.(𝒙/𝑦)^{4} = ?

Here, raise both variables within the brackets by the power of four.

Take a look at this more complicated equation:(4𝒙^{3}/5𝑦 ^{4})^{2} = ?

Don’t forget to distribute the exponent you’re multiplying by to **both** the coefficient and the variable. Then simplify where possible.(4𝒙^{3}/5𝑦^{4})^{2} = 4^{2}𝒙^{6}/5^{2}𝑦^{8} = 16𝒙^{6}/25𝑦^{8}

### 6. Zero power rule

Any base raised to the power of zero is equal to one.

The easiest way to explain this rule is by using the quotient of powers rule.

4^{3}/4^{3} = ?

Following the quotient of powers rule, subtract the exponents from each other, which cancels them out, only leaving the base. Any number divided by itself is one.

4^{3}/4^{3} = 4/4 = 1

No matter how long the equation, anything raised to the power of zero becomes one.

(8^{2}𝒙^{4}𝑦^{6})^{0} = ?

Typically, the outside exponent would have to be multiplied throughout each number and variable in the brackets. However, since this equation is being raised to the power of zero, these steps can be skipped and the answer simply becomes one.

(8^{2}𝒙^{4}𝑦^{6})^{0} = 1

The equation fully expanded would look like this:

(8^{2}𝒙^{4}𝑦^{6})^{0} = 8^{0}𝒙^{0}𝑦^{0} = (1)(1)(1) = 1

### 7. Negative exponent rule

When there is a number being raised by a negative exponent, flip it into a reciprocal to turn the exponent into a positive. **Don’t** use the negative exponent to turn the base into a negative.

We’ve talked about reciprocals before in our article, “**How to divide fractions in 3 easy steps**”. Essentially, reciprocals are what you multiply a number by to get the value of one. For example, to turn two into one, multiply it by ½.

Now, look at this exponent example:

𝒙^{-2} = ?

To make a number into a reciprocal:

- Turn the number into a fraction (put it over one)
- Flip the numerator into the denominator and vice versa
- When a negative number switches places in a fraction it becomes a positive number

The goal of equations with negative exponents is to make them positive.Now, take a look at this more complicated equation:

4𝒙^{-3}𝑦^{2}/20𝒙v^{-3} = ?

In this equation, there are two exponents with negative powers. Simplify what you can, then flip the negative exponents into their reciprocal form. In the solution, 𝒙^{-3} moves to the denominator, while 𝑧^{-3} moves to the numerator.

Since there is already an 𝒙 value in the denominator, 𝒙^{3} adds to that value.

4𝒙^{-3}𝑦^{2}/20𝒙𝑧^{-3} = 𝑦^{2}𝑧^{3}/5𝒙^{4}

With these seven rules in your students’ back pockets, they’ll be able to take on most exponent questions they come across!

reciprocal, a^{-m} = 1/a^{m}

## How Prodigy can help you teach exponent rules

Prodigy is a curriculum-aligned math game you can use to assign questions, track progress, and identify trouble spots in your students’ learning. And you can create teacher and student accounts for free!

With so many different exponent rules to follow and several students to track, it can be hard to see who needs help with what. Prodigy makes it easy to track progress, and create a unique gaming experience for each student based on their needs.

Statistics are tracked live, as students play the game, and feedback is available instantly. Most of the time your students won’t even realize that they’re taking part in math lessons. It’s all part of their personalized gaming experience!

From the teacher dashboard, you can create lesson plans, see live statistics, input custom assignments, and prepare your students for upcoming tests.

**Formulas and properties of exponents** are used in the reduction and simplification of expressions, and in solving equations and inequalities.

1. | a^{0} = 1 (a ≠ 0) | – zero exponent property |

2. | a^{1} = a | – any number raised by the exponent 1 is the number itself |

3. | a^{n} · a^{m} = a^{n + m} | – product of powers property |

4. | (a^{n})^{m} = a^{nm} | – power of a power property |

5. | a^{n}b^{n} = (ab)^{n} | – power of a product property |

6. | a^{-n} = 1a^{n} | – negative exponent property |

7. | a^{n}a^{m} = a^{n – m} | – quotient of powers property |

8. | a^{1/n} = ^{n}√a | – rational exponent property |

**WHAT IS AN EXPONENT?**

First let’s look at how to work with variables to a given power, such as a^{3}.

There are five rules for working with exponents:

1. a

^{m}* a^{n}= a^{(m+n)}2. (a * b)

^{n}= a^{n}* b^{n}3. (a

^{m})^{n}= a^{(m * n)}4. a

^{m}/ a^{n}= a^{(m-n)}5. (a/b)

^{n}= a^{n}/ b^{n}

Let’s look at each of these in detail.

1. **a ^{m} * a^{n} = a^{(m+n)}** says that when you take a number, a, multiplied by itself m times, and multiply that by the same number a multiplied by itself n times, it’s the same as taking that number a and raising it to a power equal to the sum of m + n.

Here’s an example where

a = 3

m = 4

n = 5

a^{m}* a^{n}= a^{(m+n)}

3^{4}* 3^{5}= 3^{(4+5)}= 3^{9}= 19,683

2. **(a * b) ^{n} = a^{n} * b^{n}** says that when you multiply two numbers, and then multiply that product by itself n times, it’s the same as multiplying the first number by itself n times and multiplying that by the second number multiplied by itself n times.

Let’s work out an example where

a = 3

b = 6

n = 5

(a * b)^{n}= a^{n}* b^{n}

(3 * 6)^{5}= 3^{5}* 6^{5}

18^{5}= 3^{5}* 6^{5}= 243 * 7,776 = 1,889,568

3. **(a ^{m})^{n} = a^{(m * n)} **says that when you take a number, a , and multiply it by itself m times, then multiply that product by itself n times, it’s the same as multiplying the number a by itself m * n times.

Let’s work out an example where

a = 3m = 4n = 5

(a^{m})^{n}= a^{(m * n)}

(3^{4})^{5}= 3^{(4 * 5)}= 3^{20}= 3,486,784,401

4. **a ^{m} / a^{n} = a^{(m-n)}** says that when you take a number, a, and multiply it by itself m times, then divide that product by a multiplied by itself n times, it’s the same as a multiplied by itself m-n times.

Here’s an example where**a = 3m = 4n = 5**

**a ^{m} / a^{n} = a^{(m-n)}**

**3 ^{4} / 3^{5} = 3^{(4-5)} = 3^{-1} (Remember how to raise a number to a negative exponent.)**

**3 ^{4} / 3^{5} = 1 / 3^{1} = 1/3**

5. **(a/b) ^{n} = a^{n} / b^{n}** says that when you divide a number, a by another number, b, and then multiply that quotient by itself n times, it is the same as multiplying the number by itself n times and then dividing that product by the number b multiplied by itself n times.

Let’s work out an example where

a = 3

b = 6

n = 5

(a/b)^{n}= a^{n}/ b^{n}

(3/6)^{5}= 3^{5}/ 6^{5}Remember 3/6 can be reduced to 1/2. So we have:

(1/2)^{5}= 243 / 7,776 = 0.03125

Understanding exponents will prepare you to use logarithms.

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