Squares and Cubes

Dhristi JEE 2022-24

What are Squares and Cubes?

A number's square can be calculated by multiplying it by itself. We can use multiplication tables to calculate the square for single-digit numbers, but we must multiply the number by itself to get the answer for two or more than two-digit values. To find the cube of any number, multiply it by itself first, then multiply the result by the original number once more. In simple words, we can conclude that the cube of a number N is N × N × N. 

Most of the students are confused about the functions of squares, cubes, and their roots, and it is hard to understand these numbers without memorising them. This article will be a reference tool and also provides brief notes of squares, cubes, square roots, cube roots, and square and cube tables. 


Squares

In the introduction, we already discussed that a square is a number multiplied by itself.  The 4 x 4 is 4 squared for example. The number squared is expressed as n\[^{2}\] in a mathematical notation with "n" signifying any number. The following is a list of common squares upto 50:


Squares Upto 50 Chart: 

The values of Squares upto 50 are listed in the given table below. Learning Square and Cube 1 to 50 can help students to memorise all squares and cubes up to 4 digits and helps to approximate a square root by interpolating between the known squares. At least students should be able to learn Squares and Cubes from 1 to 30.

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Square Roots

Squares and square roots are inverse operations in mathematics. A number's square can be calculated by multiplying it by itself, while the square root of a number will be the number that we need to multiply by itself to obtain the original number. For example, let’s consider that “a” is the square root of “b”, then it means that a×a will be equal to b. The square of any number will always be a positive number, so every number will have two square roots, one negative value, and one positive value. 

⇒ For Example, 2 and -2 are square roots of 4. But at most places, you will find that only the positive value is written as the square root.

In simple words, we can say that the reverse operation of squaring a number is nothing but its square root, and square roots are written by the symbol “√” over them. Because squares and square roots are opposite operations. 

⇒ For example, √25 = 5 because 5\[^{2}\] = 25.


Square Root by Prime Factorization Method

The term "prime factorization" refers to the representation of a number as a product of prime numbers. Follow the steps below to find the square root of a given number using the prime factorization method:

Step 1: Dividing the provided number into prime factors is the first step.

Step 2: Create pairs of similar factors in which both factors in each of the pairs are equal.

Step 3: Choose one of the factors.

Step 4: Take one factor from each pair and find the product of the factors.

Step 5: The square root of the given number is the product.


Cubes

Cube numbers or perfect cubes are the product of the multiplication of a number three times by itself.

Few examples of Cubes: 1, 8, 27, …, etc.

The cube of a number m will be denoted by m\[^{3}\] and it can be expressed as m\[^{3}\] = m × m × m.

Therefore, 1\[^{3}\] = 1 × 1 × 1 = 1, 2\[^{3}\] = 2 × 2 × 2 = 8, and 3\[^{3}\] = 3 × 3 × 3 = 27 are elaborated examples of cubes.

Let’s learn cubes from 1-30 from the table below. By finishing this chart will enable us to memorise Squares and Cubes from 1 to 30.

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Some Interesting Patterns

If each prime factor occurs three times in the prime factorization of any number, the number is the perfect cube.

Let’s look at an example, 

⇒ 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2\[^{3}\] × 3\[^{3}\] = (2 × 3)\[^{3}\] = 6\[^{3}\] which is nothing but a perfect cube.

 

Smallest Multiple that is a Perfect Cube

Generally, we must find the smallest natural number in which a number may be multiplied or divided to form a perfect cube.

 

Cube Roots

The cube root is simply the inverse operation of getting the cube.

For example, 2\[^{3}\] = 8 ⇒ 2 is the cube root of 8.

This \[\sqrt[3]{}\] symbol denotes the cube root. Thus, the above example can be written as \[\sqrt[3]{8}\] = 2

 

Conclusion

Squares and square roots are inverse operations in mathematics. The square root of any number is the number that needs to be multiplied by itself to get the original number, whereas the square of a number is the number that needs to be multiplied by itself. If a is the square root of b, then axa=b. Every number will have two square roots, one is a positive value and another one negative value because the square of any number is always a positive number. When we multiply a number by itself twice, we get the product of three numbers, all of which are the same. The cube of the original number is the resultant number (product). It's called a cube since it's used to indicate a cube's volume. In other terms, the cube of a number is raised to exponent 3. In this article, we have seen a brief explanation of squares, cubes, square roots, cube roots, and square, cube tables.

FAQs on Squares and Cubes

1. What are the Squares and Cubes?

Answer: A number's square can be calculated by multiplying it by itself. We can use multiplication tables to calculate the square for single-digit numbers, but we must multiply the number by itself to get the answer for two or more than two-digit values. To find the cube of any number, multiply it by itself first, then multiply the result by the original number once more. In simple words, we can say that the cube of a number N is N × N × N.  

2. How Important is the Square and Cube Chart?

Answer: Great skills on Square and Cube charts allow students to figure out complex problems of higher classes. Memorising the charts will help students to solve problems in exams easily. 

3. What are Some Methods to Square the Root of a Number?

Answer: Any of the four methods listed below can be used to find the square root of a number:

  • Repeated Subtraction Method

  • Prime Factorization Method

  • Estimation and Approximation Method

  • Long Division Method.

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