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**

**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.

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

- The number 1 raised to any power is 1 itself. \[ 1^n = 1 \]
- Any number raised to the power of 0 is 1. \[ n^0 = 1 \]
- 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.

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

- Even Exponents of Negative Numbers: For even exponents, the negative of the base is converted into a positive. \[ (-x)^{even~exponent} =x^{even~exponent} \]
- Odd Exponents of Negative Numbers: For odd exponents, the negative of the base becomes a negative for the whole expression. \[ (-x)^{odd~exponent} =-x^{odd~exponent} \]
- 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.

**About 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.

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

### 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.

\[\begin{align} \ (a^m)^n&= a^{m \times n}\\(3^2)^4&= 3^{2 \times 4} = 3^8\end{align} \]

### 9. What is the difference between exponents and powers?

Exponents and powers refer to the same number n in \(a^n \).

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

The two exponents can be compared by making the bases equal.

\[\begin{align} 2^7, 4^2,8^2 \\2^7, (2^2)^2, (2^3)^2 \\2^7, 2^4, 2^6 \end{align} \]