Exponents, Squares and Cubes
Let us begin with what you know. You can easily write 2 x 2 = 4 without much difficulty, or even 3 x 3 x 3 = 27, however, what do you do when you need to represent 2 multiplied to itself for 10 times?
That's a tough one right? Mathematicians have already faced and addressed this particular hurdle and that brings us to the topics of exponents, squares, and cubes.
Let us learn more about the exponents, squares, and cubes as we scroll down.
Lesson Plan
What Are Exponents?
An exponent is a number that indicates the number of times a number is multiplied by itself.
3 is the exponent here and is written as an index.
An exponent is a way to represent repeated multiplication of a number/variable with itself.
From the belowgiven picture, we can understand that when the value of the exponent decreases then the number it represents also gets smaller.
Definition of Exponents
An exponent is a number that is written above and to the right of another number or a mathematical expression (say, \(m + 2\)) called the base.
An exponent indicates that the base is raised to a certain power.
The exponent is also called by other names like index and power.
Definition: If \(m\) is a positive number and \(n\) is its exponent, then \(m^{n}\) means \(m\) is multiplied by itself \(n\) times.
Example:
Squares and Cubes are limiting cases of exponents where the exponent component is always 2 and 3 respectively.
As now you have understood about the exponents and what does it mean, let us jump into an interesting topic related to it.
Examples of Exponents
Some examples of exponents are as follows:
 \(\begin{align} 3 \times 3 \times 3 \times 3 \times 3 &= 3^5\end{align}\)
 \(\begin{align}2 \times 2 \times 2 &= (2)^3\end{align}\)
 \(\begin{align}a \times a \times a \times a \times a \times a &=a^6\end{align}\)
You can calculate the exponents using the following "Exponent Calculator".
Why Are Exponents Important?
Exponents are important because, without them, the products where a number is repeated by itself many times is very difficult to write.
For example, it is very easy to write \(5^7\) instead of writing \(5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5\).
Different Properties of Exponents
The properties of exponents are as follows:
The negative exponent of a number is defined as how many times we are multiplying the reciprocal of that number.
Squares and Square roots
Just like how squares represent a number or a variable multiplied by itself twice, a square root stands for the number that needs to be multiplied by itself to get that particular number.
Essentially it is the inverse operation of a square.
Note: Numbers that give integers when their square roots are calculated are called Perfect Squares.
Examples of perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.
Cubes and Cube Roots
Similar to square roots, cube roots are the inverse operations of cubes.
So, if a cube represents a number multiplied by itself thrice, the cube root of a number represents the number that is multiplied 3 times to give the original number.
Note: Numbers that give integers when their square roots are calculated are called perfect cube.
The first five positive perfect cubes are 1, 8, 27, 64, and 125
 A square root can be replaced with the exponent 1/2. For example: \(\sqrt 2 = 2^{\frac 12}\)
 A cube root can be replaced with the exponent 1/3. For example: \(\sqrt[3] 8 = 8^{\frac 13}\)

The rules of exponents are as follows:
\[\begin{align} &a^0=1&;\,\,&a^1=a\\[0.2cm]&a^m \times a^n = a^{m + n}&;\,\,&\dfrac{a^m}{a^n}= a^{mn}\\[0.2cm]&a^{m} = \dfrac{1}{a^m}&;\,\,&(a^m)^n=a^{mn}\\[0.2cm]&(ab)^m=a^mb^m&;\,\,&\left(\dfrac{a}{b}\right)^m= \dfrac{a^m}{b^m}\end{align}\]
Exponent Calculator
Here is the "Exponent Calculator."
Here, you can increase/decrease the values of the base and the exponent and can see the resultant value.
 Solve: \(\dfrac{7^3 \times {3^{4}}}{{21^{2}}}\)
 Can you find the value of \((\sqrt[3]{2})^{10}(\sqrt{5})^{4}\) ?
Solved Examples
Example 1 
Help Tim to find the value of \(x\) if \(3^{x} = 27\)
Solution
\(3^{x} = 27\) means when \(3\) is multiplied by itself for \(x\) times, it gives \(27\)
We know that \(3\times3\times3=27\)
To get 27, 3 is multiplied by itself for 3 times.
i.e., \(3^{3} = 27\)
Comparing this with the given equation: \(3^x=27\)
\(x = 3\) 
Example 2 
I am a number between 1 and 100. I am a perfect square as well as a perfect cube! I am not 1, what number am I?
Solution
The perfect squares are encircled by blue, whereas the perfect cubes are highlighted in red color in the picture below.
64 is the only number between 1 and 100 which is both a perfect square as well as a perfect cube as
\[8^{2}=64\]
\[4^{3}=64\]
\(\therefore\) The number is \(64.\) 
Example 3 
Alan is stuck with a problem. His teacher has asked him to find the volume of this cube if its base area is 729 square units. Can you help him?
Solution
Base area of the square = \(\text {side} \times \text {side} = m^{2}\)
\(m=\sqrt {\text {Base area}}\)
\(m=\sqrt {729}\)
\(m= 27\) units
Volume of the cube = \(\text {side} \times \text {side} \times \text {side} = m^{3}\)
Volume of the cube = \( 27 \times 27 \times 27 = 27^{3}\)
Volume of the cube = \( 19683\) cubic units
\(\therefore\) Volume of the cube = \( 19683\) cubic units 
Example 4 
In a forest, each tree has about \(5^7\) leaves and there are about \(5^3\) trees in the forest.
Find the total number of leaves.
Solution
The number of trees in the forest = \(5^3\)
The number of leaves in each tree = \(5^7\)
The total number of leaves are:
\[ \begin{aligned} 5^3 \times 5^7 &= 5^{10} \; [\because a^m \times a^n = a^{m+n}] \end{aligned} \]
\(\therefore\) Total number of leaves \(=5^{10}\) 
Example 5 
Here is an application problem of exponents in algebra.
The dimensions of a wardrobe are \(x^5\) in, \(y^3\) in, and \(x^8\) in. Find its volume.
Solution
The dimensions of the wardrobe are:
\[ \begin{aligned} \text{Length, } l &= x^5 \text{ in}\\[0.2in] \text{Width, } w &= y^3 \text{ in}\\[0.2in] \text{Height, } h &= x^8 \text{ in} \end{aligned} \]
The volume of the wardrobe, \(V\) is:
\[ \begin{aligned} V &= lwh\\[0.2in]&=x^5 \cdot y^3 \cdot x^8\\[0.2in]&=x^{13} \,y^3 \; [\because a^m \cdot a^n = a^{m+n} ] \end{aligned} \]
\( \therefore V =x^{13} \,y^3 \) 
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 minilesson targeted the fascinating concept of exponents, squares, and cubes. The math journey around exponents, squares, and cubes 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!
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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.
Frequently Asked Questions
1. What are exponents?
An exponent indicates that the base is raised to a certain power. The exponent is also called by other names like index and power.
Definition: If \(m\) is a positive number and \(n\) is its exponent, then \(m^{n}\) means \(m\) is multiplied by itself for \(n\) times.
2. How to calculate exponents?
Exponents can be calculated using the given rules:
\begin{aligned} a^0&=1\\[0.2cm]a^1&=a\\[0.2cm]a^m \times a^n &= a^{m + n}\\[0.2cm]\dfrac{a^m}{a^n}&= a^{mn}\\[0.2cm]a^{m} &= \dfrac{1}{a^m}\\[0.2cm](a^m)^n&=a^{mn}\\[0.2cm](ab)^m&=a^mb^m\\[0.2cm]\left(\dfrac{a}{b}\right)^m&= \dfrac{a^m}{b^m}\end{aligned}
4. How to find the square of a number?
To find a square of a number, multiply the number by itself.
For example, the square of 2 is \( 2 \times 2 = 4\)
4. How to find the cube of a number?
To find a square of a number, multiply the number with itself twice.
For example, the cube of 2 is \( 2 \times 2 \times 2 = 8\)
5. How to find square root and cube roots?
 To find the square root of a number, find the number which when multiplied by itself gives the original number.
For example: Square root of 4 is 2 because \(2 \times 2 = 4\)  To find the cube root of a number, find the number which when multiplied by itself thrice gives the original number.
For example: Cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\)