# Exponential Terms

Exponential Terms

In the below image, there are several butterflies of the same species. Now can you think of a way to write many numbers in a simplified form?

In maths, we use exponents to express many numbers in a single expression.

The number 2 multiplied 7 times to itself, can be expressed as:  $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7$

In this mini-lesson, we will learn more about exponents and how we can use them in math.

## Lesson Plan

 1 What Do You Understand by Exponential Terms? 2 Tips and Tricks 3 Important Notes on Exponential Terms 4 Solved Examples on Exponential Terms 5 Interactive Questions on Exponential Terms

## What do you Understand by Exponential Terms?

The variables having powers or index are called exponential terms.

The standard form of an exponent is $$a^n$$.

$a^n = a \times a \times a \times .... n~times$

Here 'a' is the base and 'n' is called the power, exponent, or index.

And it is read as 'a to the power of n'.

n can have values of whole numbers, integers, fractions, or decimals.

Some of the examples of exponential terms are :

$$2^5, (-4)^{0.2}, 5^{\frac{2}{3}}$$

Further, the following formulae are used to simplify exponents.

• $$a^m \times a^n = a^{m + n}$$
• $$\dfrac{a^m}{ a^n} = a^{m - n}$$
• $$(a^m)^n= a^{m \times n}$$
• $$a^{-m} = \dfrac{1}{a^m}$$
• $$\sqrt [n] a^m= (a^m)^{\frac{1}{n}} =a^{\frac{m}{n}}$$

Tips and Tricks
1. The number 1 raised to any power is 1 itself. $1^n = 1$
2. Any number raised to the power of 0 is 1. $n^0 = 1$
3. A fraction can be written as an exponent raised to the power of -1. $\dfrac{1}{n} = n^{-1}$

## How to simplify Exponential Terms raised to an exponent?

Exponential terms raised to an exponent can be conveniently transformed into simplified form.
$(a^m)^n = a^{m \times n}$

### Examples:

A few examples listed below would help us to understand the simplification of exponential terms.

• $$(2^4)^3 = 2^{4 \times 3} = 2^{12}$$
• $$(((5^2)^2)^2)^2 = 5^{2 \times 2 \times 2 \times 2} = 5^{16}$$
• $$(3^4)^5 = 3^{4 \times 5 } = 3^{20}$$

Important Notes
1. Even Exponents of Negative Numbers: For even exponents, the negative of the base is converted into a positive.  $(-x)^{even~exponent} =x^{even~exponent}$
2. Odd Exponents of Negative Numbers: For odd exponents, the negative of the base becomes a negative for the whole expression.  $(-x)^{odd~exponent} =-x^{odd~exponent}$
3. Converting Decimals into Exponents: The decimal can be converted to a negative exponent of the number 10.
$0.1 = 10^{-1}, 0.01 = 10^{-2}$
Example:$0.0035 = 35 \times 10^{-4}$

## Solved Examples

 Example 1

Simplify $$\dfrac{8}{2^{-3}}$$ to the power of 2.

Solution

\begin{align} \dfrac{8}{2^{-3}} &= \dfrac{2^3}{2^{-3}}\\&=2^3 \times 2^3 \\&=2^{3 + 3}\\&=2^6\end{align}

 $$\therefore$$ The answer is $$2^6$$
 Example 2

Find the value of $$(-2)^4 \times (\dfrac{3}{2})^4$$.

Solution

\begin{align} (-2)^4 \times (\dfrac{3}{2})^4 &= 2^4 \times \frac{3^4}{2^4}\\&=3^4 \\&=81\end{align}

 $$\therefore$$ The answer is 81.
 Example 3

If $$3^{2x - 1} \times 3^4 = 3^{11}$$  Then the value of x is________

Solution

\begin{align} 3^{2x - 1} \times 3^4 &= 3^{11} \\3^{2x - 1+4}&=3^{11}\\3^{2x+3}&=3^{11}\end{align} $\text{Bases are equal, powers can be equalized}$ \begin{align} 2x + 3 &= 11 \\2x&=11-3 \\2x&=8 \\x&=\frac{8}{2} \\x&=4\end{align}

 $$\therefore x = 4$$
 Example 4

Express 0.000064 in exponential form.

Solution

\begin{align}0.000064 &= \frac{64}{1000000}\\&=\frac{64}{10^6} \\&= 64 \times 10^6 \\&=2^6 \times 10^6\end{align}

 $$\therefore$$ The exponential form is $$2^6 \times 10^6$$
 Example 5

Simplify $$\dfrac{4^2 \times3^9 \times 5^{-4}}{2^3 \times 5^{-5} \times 9^3}$$.

Solution

\begin{align}\dfrac{4^2 \times3^9 \times 5^{-4}}{2^3 \times 5^{-5} \times 9^3} &=\dfrac{(2^2)^2 \times3^9 \times 5^{-4}}{2^3 \times 5^{-5} \times (3^2)^3}\\&=\dfrac{2^4 \times3^9 \times 5^{-4}}{2^3 \times 5^{-5} \times 3^6} \\&=2^{4 - 3} \times 3^{9 - 6} \times 5^{-4 +5}\\&=2^1 \times 3^3 \times 5^1 \\&=3^3 \times 10 \\&=27 \times 10 \\&=270\end{align}

 $$\therefore$$ The answer is 270.

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of exponential terms. The math journey around exponential terms starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## FAQs on Exponential Terms

### 1. What is an exponential term?

The term of the form $$a^n$$, having a base and power is called an exponential form.

### 2. What is an exponential example?

An example of an exponent is $$2^5$$.  $2^5 = 2 \times 2 \times 2 \times 2 \times 2$
Here 2 is called the base and 5 is called the exponent or power.

### 3. How do you classify an exponential function?

A function having a variable with powers can be classified as an exponential function. $f(x) = x^n$

### 4. What is an exponential form in maths?

$$5^3$$, $$4^{-3}$$, $$3^{\frac{1}{2}}$$ are a few examples of exponential forms in maths.

### 5. What are the five rules of exponents?

The five rules of exponents are.

1. $$a^m \times a^n = a^{m + n}$$
2. $$\dfrac{a^m}{ a^n} = a^{m - n}$$
3. $$(a^m)^n= a^{m \times n}$$
4. $$a^{-m} = \dfrac{1}{a^m}$$
5. $$\sqrt [n] a^m= (a^m)^{\frac{1}{n}} =a^{\frac{m}{n}}$$

### 6. How do you simplify exponents?

Exponents are simplified by adding or multiplying powers, for the same bases.  \begin{align}2^4 \times 2^3 &= 2^{4 + 3} = 2^7 \\ (5^3)^2 &=5^{3 \times 2} = 5^6\end{align}

### 7. What is the rule of multiplying exponents?

The rule for multiplying exponents is: \begin{align} a^m \times a^n &= a^{m + n} \\ 3^2 \times 3^5 &= 3^7\end{align}

### 8. How do you solve exponents with powers?

The terms with exponents with powers can be solved by using the following formulae.

### 10. How do you compare two exponents?

The two exponents can be compared by making the bases equal.
\begin{align} 2^7, 4^2,8^2 \\2^7, (2^2)^2, (2^3)^2 \\2^7, 2^4, 2^6 \end{align}

Exponents and Logarithms
Grade 9 | Questions Set 1
Exponents and Logarithms