Contents

# Power|Definition & Meaning

## Definition

The **power**, index, or **exponent** of a value (called the **base**) is a **number** indicating how many times that value multiplies by itself. We write the **power** of a value slightly above it to the right or using the hat (^) symbol. For example, 5 x 5 = 5^{2} = 5^2. Here, 2 is the **power** of **base** 5, and the operation (called **exponentiation**) reads as “5 to the **power** 2” or “2nd **power** of 5.”

Power and **exponents** are used to represent the combination of a **number** being multiplied by itself a fixed **number** of times. They are mostly used to demonstrate a large quantity or a very small quantity in a form that is readable and writable for the user.

For instance, we say that 3 raised to the **power** of 4 is equal to 3 multiplied by itself 4 times or 3 x 3 x 3 x 3. Instead of writing 3 several times, we can simply write it as 3^{4}, where 3 represents the **base** **number** with 4 as the **exponent**. This combination of **base** and **exponent** is known as **power**.

3 raised to **power** 4 is a simple problem to cater to, but usually big **numbers** like a light year or the distance between the sun and earth and really minor values like the mass of an electron are represented in **powers** of 10. So rather than writing a bunch of zeros, we use **power** to simplify our problems.

The **exponential number** that is placed in the upper right corner of the **base** is known as a superscript.

## What Is Power?

Power is a combination of two terms, a **base**, and an **exponent**. In mathematics, it refers to how many times a **number** is multiplied by itself. The **base** is the digit that is multiplied by itself. Whereas the **exponent** is the total **number** of times, the **base** has to be multiplied. In simple words, **power** is expressed in **exponential **form.

Furthermore, it denotes the repetition of a sequence of **multiplication **of a single **number**.

The most common **power** expression we have used is the quadratic equation, which holds terms having squared **power**. If the expression has an **exponent** of 3, it is referred to as cube **powered**. The higher **powers** are then related as ‘to the **power** of (x)’, where x is any integer.

### Exponent

An **exponent** is a numeric digit representing the repetition of a **number**. It is usually denoted by the letter n. The generic form of **power** is B^{n}, where B is the **base** with n as the **exponent**. For instance, if 8 is multiplied by itself, let’s say n times, then this means that:

8 x 8 x 8 x 8 x 8 x 8 x 8 x….. n = 8^{n}

The verbal form of the above expression is 8 raised to the **power** of n. Thus indicating that **exponents** are just **powers** or occasionally known as indices.

The general form of **exponents** is given in the following figure.

## Law of Powers

There are certain **laws** that help in simplifying expressions containing multiple **power** terms, or even a single term containing variable **power**. The laws are based on the arithmetic operations between the like and dislike **bases** and **powers**.

### Multiplication Law

When there are two terms having the same **base** but different **powers**, and they are undergoing the **arithmetic** operation of **multiplication **then this law comes in handy. It says that if the **bases** are identical, then the **powers** can add up, such that:

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

### Division Law

When there are two terms having the same **base** but different **powers**, and they are undergoing the arithmetic operation of **division **then this law comes in handy. It says that if the **bases** are identical, then the **powers** are subtracted, such that:

a^{m} / a^{n} = a^{m-n}

### Exponential Law

When there is a term having a negative **exponential **and the term is in the numerator, then this law states that the whole term undergoes a reciprocal with the negative **exponential **converting into a positive **exponential **to the **base:**

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

## Power Rules

We have gone through the different **laws** for solving **power** problems, but there are certain rules related to these laws which we are going to discuss in this topic. As of the moment, there are seven **power** rules which help us in identifying different word problems related to **powers**.

### Product Rule

Let’s say we have two similar **bases** but their **powers** are different:

4^{3} x 4^{4}

Then, we make use of the **multiplication **law and add those **powers** such that (3 + 4):

4^{3} x 4^{4} = 4^{7}

This means that 4 is multiplied by itself 7 times in a row:

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

Suppose we have a more complex situation where:

4x^{3} x 2x^{4}

Since the **base** here is x, we can add up the **exponents** and then multiply the coefficients of x:

4x^{3} x 2x^{4} = 8x^{7}

### Quotient Rule

Let’s say we have two similar **bases** but their **powers** are different in a fraction:

4^{6} / 4^{4}

Then, we make use of the **division **law and subtract those **powers** such that (6 – 4):

4^{6} x 4^{-4} = 4^{6-4} = 4^{2}

This means that 4 is multiplied by itself 2 times in a row:

4^{2} = 4 x 4 = 8

Suppose we have a more complex situation where:

4x^{2} / 2x^{1}

Since the **base** here is x, we can add up the **exponents** and then multiply the coefficients of x:

4x^{2} x 2x^{-1} = 8x^{2-1} = 8x

### Power of Power Rule

Let’s say you encounter a situation where **power** is raised to some **power** of itself:

(x^{2})^{3}

In such cases, simply multiply the **powers** together like normal integers:

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

We can break down this math:

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

(x^{2})^{3} = x.x.x.x.x.x

(x^{2})^{3} = x^{6}

### Power of Product Rule

Let’s say we have a **base** with two different variables having a single **power** outside the round brackets, such that:

(xy)^{3}

Then, we make use of the distribution and **multiplication **law and distribute those **powers** such that:

(xy)^{3} = x^{3}y^{3}

This means that x and y are multiplied by themselves 3 times individually:

x^{3}y^{3} = x.x.x.y.y.y

Suppose we have a more complex situation where:

(x^{2}y^{2})^{3}

Since there is **power** inside a **power**, we can multiply and distribute the **exponents:**

(x^{2}y^{2})^{3} = x^{6}y^{6}

### Power of Quotient Rule

Let’s say we have a **base** with two different variables in fractional form having a single **power** outside the round **brackets**, such that:

(x/y)^{3}

Then, we make use of the distribution and **division **law and distribute those **powers** such that:

(x/y)^{3} = x^{3}/y^{3}

This means that x and y are **multiplied** by themselves 3 times individually:

x^{3}/y^{3} = x.x.x./y.y.y

Suppose we have a more complex situation where:

(x^{2}/y^{2})^{3}

Since there is **power** inside a **power**, we can **multiply** and distribute the **exponents:**

(x^{2}/y^{2})^{3} = x^{6}/y^{6}

### Zero Rule

Anything raised to the **power** of zero, is 1, no matter how **complex** the equation is it will always be equated to 1.

(3^{2}.2^{5}.9^{100})^{0} = 1

## Solved Example

### Example 1

Simplify $\dfrac{x^4.y^5.z^8}{x^{-2}.y^2.z^3}$

### Solution

Using **the division **rule to solve this problem:

= $\dfrac{x^4.y^5.z^8}{x^{-2}.y^2.z^3}$

= x^{4-(-2)}.y^{5-2}.z^{8-3}

= x^{4+2}.y^{3}.z^{5}

= x^{6}.y^{3}.z^{5}

This is the **simplified** form of the **equation**.

### Example 2

Simplify 2ab + 4b (b^{2} – 2a).

### Solution

First, **reduce** the brackets:

= 2ab + 4b(b^{2}) – 4b(2a)

= 2ab + 4(b^{1+2}) – 4b(2a)

= 2ab + 4b^{3} – 8ab

= 4b^{3} – 6ab

This is the final **answer.**

*All images were created with GeoGebra.*