Squares And Square Roots Cubes And Cube Roots - Class 8 Maths

Understanding Squares and Perfect Squares

A square number or a perfect square is the result of multiplying an integer by itself. For example, when you multiply 3 by 3, you get 9. So, 9 is the square of 3. We write this as $3^2 = 9$. Similarly, $5^2 = 5 \times 5 = 25$. Here, 25 is a perfect square. Perfect squares have interesting properties: they always end with 0, 1, 4, 5, 6, or 9 at the unit's place. A number ending in 2, 3, 7, or 8 can never be a perfect square. Also, if a number has an even number of zeroes at the end, it might be a perfect square (e.g., 100, 400). If it has an odd number of zeroes (e.g., 10, 1000), it's never a perfect square. Understanding these properties can help you quickly identify if a number is a perfect square without needing to calculate its root. For instance, 122 is not a perfect square because its unit digit is 2. Knowing these simple rules can save you a lot of time in competitive exams!

How to Find Square Roots Using Prime Factorization

1. Step 1: Find Prime Factors — Start by finding all the prime factors of the given number. Prime factorization means expressing the number as a product of its prime numbers. For example, for 36, the prime factors are $2 \times 2 \times 3 \times 3$.
2. Step 2: Group Factors in Pairs — Once you have the prime factors, group them into pairs. Each pair must consist of two identical prime factors. For 36, we have $(2 \times 2) \times (3 \times 3)$.
3. Step 3: Take One Factor from Each Pair — From each group of identical prime factors, take out one factor. In our example, from $(2 \times 2)$, take out 2. From $(3 \times 3)$, take out 3.
4. Step 4: Multiply the Taken Factors — Multiply the factors you took out from each pair. This product will be the square root of the original number. For 36, $2 \times 3 = 6$. So, the square root of 36 is 6. Remember, the symbol for square root is $\sqrt{\ \ }$. So, $\sqrt{36} = 6$.

Exploring Cubes and Perfect Cubes

Similar to squares, a cube number or a perfect cube is the result of multiplying an integer by itself three times. For example, if you multiply 2 by 2 by 2, you get 8. So, 8 is the cube of 2. We write this as $2^3 = 8$. Similarly, $4^3 = 4 \times 4 \times 4 = 64$. Here, 64 is a perfect cube. Unlike perfect squares, the unit digit of a perfect cube can be any digit from 0 to 9. However, there's a unique pattern: the unit digit of the cube of a number is the same as the unit digit of the cube of its unit digit. For example, $1^3=1$, $2^3=8$, $3^3=27$ (ends in 7), $4^3=64$ (ends in 4), $5^3=125$ (ends in 5), $6^3=216$ (ends in 6), $7^3=343$ (ends in 3), $8^3=512$ (ends in 2), $9^3=729$ (ends in 9), and $10^3=1000$ (ends in 0). Notice that only 2 and 8, and 3 and 7 swap their unit digits when cubed. This property can be useful for estimating cube roots. Perfect cubes also follow a rule regarding zeroes: they must have a number of zeroes at the end that is a multiple of 3 (e.g., 1000, 8000000).

Finding Cube Roots Using Prime Factorization

1. Step 1: Determine Prime Factors — Begin by finding the prime factorization of the given number, just as you would for square roots. For example, if we want to find the cube root of 216, its prime factors are $2 \times 2 \times 2 \times 3 \times 3 \times 3$.
2. Step 2: Group Factors in Triplets — Now, instead of pairs, group identical prime factors into triplets (groups of three). For 216, we have $(2 \times 2 \times 2) \times (3 \times 3 \times 3)$.
3. Step 3: Take One Factor from Each Triplet — From each group of three identical prime factors, take out one factor. From $(2 \times 2 \times 2)$, take out 2. From $(3 \times 3 \times 3)$, take out 3.
4. Step 4: Multiply the Taken Factors — Multiply these selected factors together. The product is the cube root of the original number. For 216, $2 \times 3 = 6$. So, the cube root of 216 is 6. The symbol for cube root is $\sqrt[3]{\ \ }$. So, $\sqrt[3]{216} = 6$.

Quick Tips for Squares, Square Roots, Cubes, and Cube Roots

To excel in this chapter, remember these important points:

1. Unit Digit Check: For perfect squares, the unit digit can only be 0, 1, 4, 5, 6, or 9. If a number ends in 2, 3, 7, or 8, it's NOT a perfect square. For perfect cubes, the unit digit can be any digit, but its pattern is predictable.
2. Number of Zeroes: A perfect square must have an even number of zeroes at its end (e.g., 100, 400). A perfect cube must have a number of zeroes that is a multiple of 3 (e.g., 1000, 8000).
3. Prime Factorization is Key: For both square roots and cube roots, the prime factorization method is reliable. Always ensure you group factors correctly – in pairs for square roots and in triplets for cube roots.
4. Memorize Small Cubes: Knowing the cubes of numbers from 1 to 10 ($1^3=1, 2^3=8, \dots, 10^3=1000$) can significantly speed up calculations and help with estimations, especially for cube roots.

Practice Questions with Solutions

• Q: Is 256 a perfect square? If yes, find its square root using prime factorization.
  A: Step 1: Find the prime factors of 256. $256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$.
  Step 2: Group the factors in pairs. $(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)$.
  Step 3: Take one factor from each pair. $2 \times 2 \times 2 \times 2$.
  Step 4: Multiply these factors. $2 \times 2 \times 2 \times 2 = 16$.
  Final answer: Yes, 256 is a perfect square, and its square root is 16.

• Q: Find the smallest number by which 392 must be multiplied so that the product is a perfect cube.
  A: Step 1: Find the prime factors of 392. $392 = 2 \times 2 \times 2 \times 7 \times 7$.
  Step 2: Group the factors into triplets. We have one triplet of 2s: $(2 \times 2 \times 2)$. The factor 7 appears twice: $(7 \times 7)$. To make it a triplet, we need one more 7.
  Step 3: The smallest number to multiply by is 7.
  Final answer: 392 must be multiplied by 7 to make it a perfect cube ($392 \times 7 = 2744 = 14^3$).

• Q: What is the cube root of 1728?
  A: Step 1: Find the prime factors of 1728. $1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$.
  Step 2: Group the factors into triplets. $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3)$.
  Step 3: Take one factor from each triplet. $2 \times 2 \times 3$.
  Step 4: Multiply these factors. $2 \times 2 \times 3 = 12$.
  Final answer: The cube root of 1728 is 12.

• Q: A perfect square number always ends with which digits? Can a number ending with 7 be a perfect square?
  A: Step 1: Recall the properties of perfect squares. Perfect squares always end with 0, 1, 4, 5, 6, or 9 at the unit's place.
  Step 2: Check if 7 is in this list of allowed unit digits.
  Final answer: A perfect square number always ends with 0, 1, 4, 5, 6, or 9. No, a number ending with 7 can never be a perfect square.

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