SQUARES AND SQUARE ROOTS, CUBES AND CUBE ROOTS

Brief Overview

graph TD; A[Number Operations] --> B[Squares n²]; A --> C[Square Roots √n]; A --> D[Cubes n³]; A --> E[Cube Roots ∛n];

Introduction

This topic covers the fundamental operations of squaring and cubing numbers, as well as finding their respective roots. These concepts are essential for higher-level mathematics, including algebra and geometry.

1. Geometrical pattern of square numbers, squares of decimals and fractions

Geometrical Pattern: A square number ($n^2$) can be represented geometrically as a square array of dots. For example, $3^2 = 9$ can be drawn as a $3 \times 3$ grid of dots. The sum of the first $n$ odd natural numbers equals $n^2$ (e.g., $1+3+5 = 9 = 3^2$).
Decimals: To square a decimal, square the number ignoring the decimal point, then place the decimal point so there are twice as many decimal places as in the original number. (e.g., $0.5^2 = 0.25$).
Fractions: To square a fraction, square the numerator and the denominator separately. $(\frac{a}{b})^2 = \frac{a^2}{b^2}$.

MCQs - Geometrical pattern, decimals and fractions

The sum of the first 4 odd natural numbers ($1+3+5+7$) is equal to:
a) $4^2$
b) $5^2$
c) $3^2$
d) $2^2$
Answer: a
What is the square of $0.12$?
a) $0.144$
b) $1.44$
c) $0.0144$
d) $0.00144$
Answer: c
The square of the fraction $\frac{3}{7}$ is:
a) $\frac{6}{14}$
b) $\frac{9}{49}$
c) $\frac{9}{14}$
d) $\frac{6}{49}$
Answer: b
A square number can always be geometrically represented as a:
a) Triangle of dots
b) Line of dots
c) Square grid of dots
d) Circle of dots
Answer: c
If $x = 1.5$, then $x^2$ is:
a) $2.25$
b) $22.5$
c) $0.225$
d) $3.0$
Answer: a
The sum of the first 10 odd natural numbers equals:
a) 20
b) 50
c) 100
d) 200
Answer: c
What is the square of the decimal $0.04$?
a) $0.16$
b) $0.016$
c) $0.0016$
d) $1.6$
Answer: c
When a fraction between $0$ and $1$ is squared, the result is:
a) Larger than the original fraction
b) Smaller than the original fraction
c) Equal to the original fraction
d) A negative number
Answer: b
The geometric pattern of a square number $16$ can be represented as a grid of dimensions:
a) $2 \times 8$
b) $4 \times 4$
c) $8 \times 2$
d) $16 \times 1$
Answer: b
The square of $\frac{5}{8}$ is:
a) $\frac{10}{16}$
b) $\frac{25}{64}$
c) $\frac{25}{8}$
d) $\frac{10}{64}$
Answer: b

2. Determination of square root of perfect square root

Definition: The square root of a number $x$ is a number $y$ such that $y^2 = x$. It is denoted by $\sqrt{x}$.
A perfect square is an integer that is the square of an integer (e.g., $16 = 4^2$).
Every positive perfect square has two square roots: one positive and one negative (e.g., $\sqrt{25} = 5$ or $-5$). Usually, the principal (positive) root is considered.

MCQs - Determination of square root of perfect square root

The principal square root of $64$ is:
a) $8$
b) $-8$
c) $32$
d) $128$
Answer: a
Which of the following is NOT a perfect square?
a) $121$
b) $144$
c) $200$
d) $225$
Answer: c
If $y^2 = 81$, the possible values of $y$ are:
a) $9$ only
b) $-9$ only
c) $9$ and $-9$
d) $81$ and $-81$
Answer: c
The square root of $10000$ is:
a) $10$
b) $100$
c) $1000$
d) $5000$
Answer: b
The square root of a positive perfect square is always:
a) An integer (if considering the principal root)
b) A fraction
c) An irrational number
d) Zero
Answer: a
If the area of a square field is $144 \text{ m}^2$, what is the length of one side?
a) $12 \text{ m}$
b) $24 \text{ m}$
c) $36 \text{ m}$
d) $72 \text{ m}$
Answer: a
Which of the following numbers has a principal square root of 15?
a) 125
b) 150
c) 225
d) 325
Answer: c
What is the value of $\sqrt{16} + \sqrt{9}$?
a) $\sqrt{25}$
b) $5$
c) $7$
d) $12$
Answer: c
The square root of $0.0009$ is:
a) $0.003$
b) $0.03$
c) $0.3$
d) $3.0$
Answer: b
If $x^2 = y^2$ and $x \neq y$, then:
a) $x$ and $y$ are both $0$
b) $x = -y$
c) $x$ and $y$ are positive
d) This is impossible
Answer: b

3. Estimation of square root of a non-perfect square

For non-perfect squares (like $10$, $50$), the square root is an irrational number (a non-repeating, non-terminating decimal).
Estimation: Find the closest perfect squares below and above the number. For example, to estimate $\sqrt{30}$:
- $25 < 30 < 36$
- $5^2 < 30 < 6^2$
- So, $\sqrt{30}$ is between 5 and 6. Since $30$ is exactly halfway between $25$ and $36$, $\sqrt{30} \approx 5.5$.

MCQs - Estimation of square root of a non-perfect square

The value of $\sqrt{40}$ lies between which two consecutive integers?
a) 5 and 6
b) 6 and 7
c) 7 and 8
d) 39 and 41
Answer: b
The best integer estimate for $\sqrt{85}$ is:
a) 8
b) 9
c) 10
d) 11
Answer: b
$\sqrt{150}$ is between:
a) 10 and 11
b) 11 and 12
c) 12 and 13
d) 14 and 15
Answer: c
Which of the following is an irrational number?
a) $\sqrt{49}$
b) $\sqrt{0.25}$
c) $\sqrt{20}$
d) $\sqrt{10000}$
Answer: c
If $\sqrt{x}$ is between 8 and 9, then $x$ could be:
a) 60
b) 75
c) 85
d) 100
Answer: b
The value of $\sqrt{10}$ is closest to:
a) $3.1$
b) $3.5$
c) $3.9$
d) $4.1$
Answer: a
Between which two integers does $\sqrt{120}$ lie?
a) 9 and 10
b) 10 and 11
c) 11 and 12
d) 12 and 13
Answer: b
Which integer is the best estimate for $\sqrt{200}$?
a) 13
b) 14
c) 15
d) 16
Answer: b
If you know that $4^2 = 16$ and $5^2 = 25$, a reasonable estimate for $\sqrt{18}$ is:
a) $4.2$
b) $4.5$
c) $4.8$
d) $5.1$
Answer: a
The irrational number $\sqrt{50}$ is bounded by the perfect squares:
a) 36 and 49
b) 49 and 64
c) 64 and 81
d) 25 and 36
Answer: b

4. Cubes and cube roots

Cube: The cube of a number is the number multiplied by itself twice ($n^3 = n \times n \times n$). Example: $4^3 = 4 \times 4 \times 4 = 64$.
Cube Root: The cube root of a number $x$ is a number $y$ such that $y^3 = x$, denoted by $\sqrt[3]{x}$. Example: $\sqrt[3]{125} = 5$.
Unlike square roots, the cube root of a negative number is negative (e.g., $\sqrt[3]{-8} = -2$).

MCQs - Cubes and cube roots

The cube of 6 is:
a) 36
b) 216
c) 18
d) 1296
Answer: b
The cube root of $-27$ is:
a) 3
b) $-3$
c) 9
d) Not a real number
Answer: b
If $x^3 = 343$, then $x$ is:
a) 7
b) 9
c) 13
d) 17
Answer: a
The cube of $0.2$ is:
a) $0.08$
b) $0.8$
c) $0.008$
d) $0.0008$
Answer: c
Which of the following is a perfect cube?
a) 100
b) 512
c) 900
d) 1024
Answer: b
The cube of a negative number is always:
a) Positive
b) Negative
c) Zero
d) Complex
Answer: b
What is the value of $\sqrt[3]{1000} - \sqrt[3]{125}$?
a) $5$
b) $10$
c) $15$
d) $75$
Answer: a
The volume of a perfect cube is $64 \text{ cm}^3$. Its side length is:
a) $2 \text{ cm}$
b) $4 \text{ cm}$
c) $8 \text{ cm}$
d) $16 \text{ cm}$
Answer: b
The cube of $1.1$ is:
a) $1.21$
b) $1.331$
c) $13.31$
d) $1.111$
Answer: b
$\sqrt[3]{-64}$ evaluates to:
a) $4$
b) $-4$
c) $8$
d) Not a real number
Answer: b

5. Finding cube root by factor method

Method: Find the prime factorization of the number. Group the identical prime factors in triplets (groups of three). Take one factor from each triplet and multiply them.
Example: $\sqrt[3]{216}$.
- $216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$
- Grouped: $(2 \times 2 \times 2) \times (3 \times 3 \times 3)$
- Cube root = $2 \times 3 = 6$.

MCQs - Finding cube root by factor method

The prime factorization of $1000$ for finding its cube root is grouped as:
a) $(2\times2\times2) \times (5\times5\times5)$
b) $(2\times5) \times (2\times5)$
c) $(10\times10\times10)$
d) $(4\times250)$
Answer: a
If the prime factorization of a number is $2^6 \times 3^3$, its cube root is:
a) $2^2 \times 3^1 = 12$
b) $2^3 \times 3^1 = 24$
c) $2^1 \times 3^1 = 6$
d) $2^6 \times 3^3$
Answer: a
To find $\sqrt[3]{-3375}$, we first find the cube root of $3375$. The prime factors of $3375$ are triplets of:
a) 3 and 7
b) 3 and 5
c) 5 and 7
d) 2 and 5
Answer: b (Cube root is $15$, so $-15$)
If a number is a perfect cube, every prime factor in its factorization must have an exponent that is a multiple of:
a) 2
b) 3
c) 4
d) 5
Answer: b
What is the cube root of $8000$?
a) 20
b) 40
c) 200
d) 400
Answer: a
If the prime factorization of $x$ is $2^9 \times 5^3$, what is $\sqrt[3]{x}$?
a) $2^3 \times 5 = 40$
b) $2^3 \times 5^1 = 40$
c) $2^6 \times 5^2$
d) $2^9 \times 5$
Answer: a
To find $\sqrt[3]{1728}$, we can group its prime factors into triplets of:
a) Twos only
b) Threes only
c) Twos and Threes
d) Twos and Fives
Answer: c
A number is multiplied by its square. The result has prime factors grouped strictly in triplets. This implies the original number was:
a) A perfect square
b) A prime number
c) Any integer
d) A fraction
Answer: c
What is the cube root of $216000$?
a) 60
b) 120
c) 600
d) 3600
Answer: a
If $y^3 = 2^3 \times 3^6$, what is the value of $y$?
a) 6
b) 12
c) 18
d) 54
Answer: c

6. Identification of square numbers

Properties:
1. A number ending in 2, 3, 7, or 8 is NEVER a perfect square.
2. A perfect square always ends with 0, 1, 4, 5, 6, or 9.
3. A number ending with an odd number of zeroes is never a perfect square.
4. The digital root (sum of digits until a single digit is reached) of a perfect square is always 1, 4, 7, or 9.

MCQs - Identification of square numbers

Which of the following numbers cannot be a perfect square?
a) 1024
b) 2025
c) 3042
d) 4096
Answer: c (Ends in 2)
A perfect square can end with which of the following digits?
a) 3
b) 7
c) 8
d) 6
Answer: d
Which of the following is a perfect square?
a) 1000
b) 40000
c) 900000
d) 250
Answer: b (Even number of zeros)
If a number ends in 5, its square will end in:
a) 0
b) 5
c) 25
d) Both b and c are correct in context
Answer: d
The square of an even number is always:
a) Odd
b) Even
c) Prime
d) A multiple of 10
Answer: b
A number ending with the digit 3 can never be a:
a) Prime number
b) Perfect square
c) Odd number
d) Perfect cube
Answer: b
Which of the following numbers could possibly be a perfect square?
a) 542
b) 987
c) 144
d) 308
Answer: c (Ends in 4, others end in 2, 7, 8)
A perfect square cannot end with which of the following numbers of zeroes?
a) 2
b) 4
c) 3
d) 6
Answer: c (Odd number of zeroes)
The square of an odd number is always:
a) Odd
b) Even
c) Prime
d) A multiple of 5
Answer: a
If the digital root of a number is 2, it:
a) Is definitely a perfect square
b) Cannot be a perfect square
c) Might be a perfect square
d) Is definitely a perfect cube
Answer: b (Perfect squares have digital roots 1, 4, 7, 9)

7. Finding square root of perfect square numbers by factor method

Method: Similar to the cube root method, find the prime factorization. Group identical factors in pairs (groups of two). Take one factor from each pair and multiply.
Example: $\sqrt{144}$.
- $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3$
- Pairs: $(2 \times 2) \times (2 \times 2) \times (3 \times 3)$
- Square root = $2 \times 2 \times 3 = 12$.

MCQs - Finding square root by factor method

To find the square root of $324$ using prime factorization, the factors are grouped in:
a) Threes
b) Fours
c) Pairs (Twos)
d) Fives
Answer: c
The prime factorization of $400$ is $2^4 \times 5^2$. Its square root is:
a) $2^2 \times 5 = 20$
b) $2 \times 5 = 10$
c) $2^4 \times 5 = 80$
d) $2^2 \times 5^2 = 100$
Answer: a
If $x = 2 \times 2 \times 7 \times 7$, then $\sqrt{x}$ is:
a) 14
b) 28
c) 98
d) 196
Answer: a
Which of the following numbers cannot be resolved into pairs of identical prime factors?
a) 256
b) 625
c) 128
d) 900
Answer: c (128 is $2^7$, which has an odd number of factors)
The square root of $1764$ is:
a) 32
b) 42
c) 52
d) 62
Answer: b (Factors are $2^2 \times 3^2 \times 7^2$)
By the prime factorization method, if $n = p^4 \times q^2$, where $p$ and $q$ are prime, then $\sqrt{n}$ is:
a) $p^2 \times q$
b) $p \times q$
c) $p^4 \times q$
d) $p^2 \times q^2$
Answer: a
What is the square root of $900$ using prime factorization?
a) $20$
b) $30$
c) $40$
d) $90$
Answer: b
A number is formed by multiplying exactly three different prime numbers together. Can it be a perfect square?
a) Yes, always
b) No, never
c) Yes, if they are all odd
d) Only if 2 is one of the primes
Answer: b (All prime factors must be in pairs)
The prime factorization of a perfect square will always have exponents that are:
a) Odd
b) Even
c) Prime
d) Negative
Answer: b
What is $\sqrt{225}$?
a) $15$
b) $25$
c) $35$
d) $45$
Answer: a

8. Finding square of perfect square numbers, decimals and numbers which are not perfect

(Note: The syllabus wording implies finding squares of various types of numbers).
* Standard Method: Multiply the number by itself.
* Using Identities: $(a+b)^2 = a^2 + 2ab + b^2$.
* Example: $103^2 = (100 + 3)^2 = 10000 + 2(100)(3) + 9 = 10609$.
* Squaring numbers ending in 5: For a number like $n5$, the square ends in $25$, and the prefix is $n \times (n+1)$.
* Example: $65^2 \rightarrow 6 \times 7 = 42$, append $25 \rightarrow 4225$.

MCQs - Finding squares

Using an identity, $51^2$ can be easily calculated as:
a) $(50+1)^2 = 2500 + 100 + 1$
b) $(50+1)^2 = 2500 + 1$
c) $(51+0)^2 = 2500 + 51$
d) $(60-9)^2 = 3600 - 81$
Answer: a
What is $51^2$?
a) 2501
b) 2601
c) 2551
d) 2611
Answer: b
Using the trick for numbers ending in 5, what is $85^2$?
a) 6425
b) 7225
c) 8025
d) 8525
Answer: b (Because $8 \times 9 = 72$, append 25)
The square of $1.2$ is:
a) $1.44$
b) $14.4$
c) $0.144$
d) $2.4$
Answer: a
If you square an irrational number like $\sqrt{5}$, the result is:
a) An irrational number
b) A perfect square integer
c) The integer 5
d) Zero
Answer: c
Using the identity $(a-b)^2 = a^2 - 2ab + b^2$, $99^2$ is:
a) $10000 - 100 + 1$
b) $10000 - 200 + 1$
c) $9800 + 1$
d) $10000 - 1$
Answer: b ($9801$)
What is the square of $35$ using the trick for numbers ending in 5?
a) 1225
b) 1125
c) 925
d) 1525
Answer: a ($3 \times 4 = 12$, append 25)
The square of $105$ is:
a) 10025
b) 11025
c) 11525
d) 10525
Answer: b ($10 \times 11 = 110$, append 25)
The square of $-12$ is:
a) $-144$
b) $144$
c) $24$
d) $-24$
Answer: b
Using $(a+b)^2$, $101^2$ evaluates to:
a) $10201$
b) $10101$
c) $10001$
d) $10211$
Answer: a

9. Finding smallest number to be added/subtracted to make a perfect square

Subtraction: Use the long division method for square roots. The remainder obtained is the smallest number that must be subtracted from the given number to make it a perfect square.
Addition: Find the square root by long division. Take the next integer greater than the quotient, square it, and subtract the original number from this new square.

Example (Subtraction): Make 130 a perfect square.
Closest perfect square below 130 is 121 ($11^2$). $130 - 121 = 9$. Subtract 9.
Example (Addition): Make 130 a perfect square.
Closest perfect square above 130 is 144 ($12^2$). $144 - 130 = 14$. Add 14.

MCQs - Smallest number to add/subtract

What is the smallest number to be subtracted from 50 to make it a perfect square?
a) 1
b) 2
c) 14
d) 25
Answer: a ($50 - 1 = 49 = 7^2$)
What is the smallest number to be added to 60 to make it a perfect square?
a) 4
b) 5
c) 21
d) 81
Answer: a ($60 + 4 = 64 = 8^2$)
What must be subtracted from 260 to get a perfect square?
a) 2
b) 4
c) 5
d) 35
Answer: b ($260 - 4 = 256 = 16^2$)
If a number $n$ lies between $a^2$ and $(a+1)^2$, what must be added to $n$ to make it a perfect square?
a) $n - a^2$
b) $(a+1)^2 - n$
c) $a^2 - n$
d) $a$
Answer: b
To make 1000 a perfect square, the smallest number to add is:
a) 0
b) 10
c) 24
d) 89
Answer: c ($31^2 = 961, 32^2 = 1024$. $1024 - 1000 = 24$)
What is the smallest integer to subtract from $20$ to get a perfect square?
a) 2
b) 4
c) 5
d) 1
Answer: b ($20 - 4 = 16$)
What is the smallest integer to add to $120$ to make it a perfect square?
a) 1
b) 5
c) 12
d) 24
Answer: a ($120 + 1 = 121$)
If a number is just 1 less than a perfect square, what must be added to make it the next perfect square?
a) 1
b) Depends on the number
c) An odd number
d) An even number
Answer: b (Actually, if $n = x^2 - 1$, adding 1 makes it $x^2$. To make it $(x+1)^2$, we add $2x+2$)
To make 99 a perfect square, what is the minimum positive number to subtract?
a) 18
b) 10
c) 9
d) 0
Answer: a ($99 - 18 = 81$)
To make 500 a perfect square, what is the smallest positive number to subtract?
a) 16
b) 24
c) 36
d) 4
Answer: a ($22^2 = 484$, so $500 - 16 = 484$)

10. Division method for finding square roots

Method:
1. Group the digits in pairs from right to left (place a bar over them).
2. Find the largest number whose square is less than or equal to the first pair. This is the first digit of the quotient.
3. Subtract the square and bring down the next pair.
4. Double the quotient to form the partial divisor, append a digit $x$ such that the new divisor times $x$ is $\le$ the new dividend.
5. Repeat until the remainder is 0 (or to the desired decimal places).

MCQs - Division method

When finding the square root of 529 by division method, the digits are paired as:
a) $\overline{52} \ \overline{9}$
b) $\overline{5} \ \overline{29}$
c) $\overline{529}$
d) $\overline{5} \ \overline{2} \ \overline{9}$
Answer: b
The division method is particularly useful for finding the square root of:
a) Very small numbers only
b) Large numbers and non-perfect squares (decimals)
c) Prime numbers only
d) Negative numbers
Answer: b
In the division method, after bringing down a pair, the new divisor is formed by:
a) Squaring the current quotient
b) Adding 10 to the current quotient
c) Doubling the current quotient and appending a new digit
d) Halving the current quotient
Answer: c
What is the square root of $529$?
a) 17
b) 23
c) 27
d) 33
Answer: b
Using the division method, what is the first digit of the square root of $10404$?
a) 1
b) 2
c) 3
d) 4
Answer: a (Pairs are $\overline{1} \ \overline{04} \ \overline{04}$, first digit is 1)
When using the division method for $\sqrt{729}$, how are the digits grouped?
a) $\overline{72} \ \overline{9}$
b) $\overline{7} \ \overline{29}$
c) $\overline{729}$
d) $\overline{7} \ \overline{2} \ \overline{9}$
Answer: b
The division method is essentially a systematic algorithm for calculating:
a) Geometric progression
b) Square roots
c) Cube roots
d) Prime factorization
Answer: b
In the division method to find $\sqrt{144}$, the first digit of the quotient is:
a) $1$
b) $2$
c) $3$
d) $4$
Answer: a
To find the square root of $0.0016$ using the division method, the first significant digit of the root will be in which place value?
a) Tenths
b) Hundredths
c) Thousandths
d) Ten-thousandths
Answer: b (The root is $0.04$)
The division method can be used to find the square root of non-perfect squares to any desired number of decimal places by:
a) Multiplying by 100
b) Appending pairs of zeroes after the decimal point
c) Guessing and checking
d) Subtracting the remainder
Answer: b

Fun Facts about Squares and Cubes!

Pythagorean Triples: Sets of three integers like $(3, 4, 5)$ where $3^2 + 4^2 = 5^2$ have been known since ancient Babylonian times, long before Pythagoras was born!
The Magic of 9: If you take any multiple of 9 and square it, the sum of the digits of the result will always eventually reduce to 9!
Fermat's Last Theorem: While $a^2 + b^2 = c^2$ has infinite integer solutions, $a^n + b^n = c^n$ has NO integer solutions for any $n > 2$ (like cubes, $a^3+b^3=c^3$). This took over 350 years to prove!
Taxicab Number: $1729$ is the smallest number that can be expressed as the sum of two cubes in two different ways ($1^3 + 12^3$ and $9^3 + 10^3$). It's famous because of a conversation between mathematicians G.H. Hardy and Srinivasa Ramanujan.

Squares And Cubes