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#6 Compulsory Number System & Arithmetic Math 6

Square Roots: Prime Factorization & Long Divisionवर्ग एवं वर्गमूल: गुणनखण्ड व भाग विधि

Mastery of square roots via prime factorization and long division methods is foundational for scoring high in the Super TET Junior Mathematics section. This module covers exact root extraction, decimal square roots, word problem modeling, and negative marking avoidance strategies aligned with SCERT UP and NCERT standards.

Preparation Status

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Syllabus Focus & High-Yield Strategy: Mastery of square roots via prime factorization and long division methods is foundational for scoring high in the Super TET Junior Mathematics section. This module covers exact root extraction, decimal square roots, word problem modeling, and negative marking avoidance strategies aligned with SCERT UP and NCERT standards.

1. Fundamental Axioms, Definitions & Foundational Properties

Definition of Square Root: If $x^2 = y$ where $x, y \in \mathbb{R}^+$, then $x$ is called the square root of $y$, denoted as $\sqrt{y} = x$. The radical sign $\sqrt{\phantom{x}}$ denotes the principal (non-negative) square root.
Multiplicative Property of Radicals: For any non-negative real numbers $a$ and $b$, $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. This fundamental axiom enables the simplification of large composite numbers during prime factorization.
Divisive Property of Radicals: For $a \ge 0$ and $b > 0$, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. This property is essential when calculating square roots of fractions and terminating rational numbers.
Unit Digit Property: A perfect square can only end in digits $0, 1, 4, 5, 6,$ or $9$ in base $10$. Numbers ending in $2, 3, 7,$ or $8$ can never be integers when square rooted, which acts as a preliminary check to eliminate incorrect options.
Exam Pro-Tip: Always check the units digit before performing long division. If a number ends in $2, 3, 7,$ or $8$, its square root is irrational or non-terminating decimal unless modified by algebraic fractions in the question.
Memory Mnemonic: SQUARE — Sign positive, Question units, Unify factors, Always pair, Root exact, Examine remainder.

2. Formulas, Standard Theorems & Algorithmic Procedures

Category / CaseFormula / Mathematical RuleTopper Calculation Shortcut
Prime Factorization Method$N = p_1^{2a} \cdot p_2^{2b} \implies \sqrt{N} = p_1^a \cdot p_2^b$Group identical prime factors in pairs of two; take one factor from each pair.
Long Division MethodPair digits from right to left (integer part) and left to right (decimal part).Find largest $y$ such that $y \times y \le ext{current group}$; double divisor for next step.
Prime Factorization Algorithm: Express the composite number $N$ as a product of prime powers: $N = 2^{a} \cdot 3^{b} \cdot 5^{c} \dots$. Ensure all exponents are even numbers. If an exponent is odd, the number is not a perfect square, requiring multiplication or division by the deficient prime to make it a perfect square.
Long Division Algorithm (Integer Part): Group digits in pairs from the decimal point moving leftwards (e.g., $1024 \to 10\;24$). Find the greatest integer $x$ whose square is less than or equal to the leftmost pair ($3^2 = 9 \le 10$). Subtract, bring down the next pair, double the quotient for the trial divisor $2x$, and find digit $y$ such that $(20x + y)y \le ext{dividend}$.
Long Division Algorithm (Decimal Part): For decimal numbers like $\sqrt{5.184}$, group digits in pairs after the decimal point from left to right, appending zeros if necessary ($5.18\;40$). Place the decimal point in the quotient exactly when the decimal point in the dividend is crossed.
15-Second Speed-Math Trick: To find the square root of a 4-digit perfect square like $7569$, look at the last digit ($9 \implies 3 \text{ or } 7$). Look at the leading block ($75 o 8^2 = 64 < 75 < 81 o 8$). Test $83 \times 83$ vs $87 \times 87$. Since $85^2 = 7225 < 7569$, the answer is $87$.

3. Word Problem Modeling & Practical Applications

Arrangement in Rows and Columns: If $N$ students or plants are arranged such that the number of rows equals the number of columns, the total number of items is $x^2$. To find items per row, calculate $\sqrt{N}$. If items are left over, subtract the remainder first before taking the square root.
Minimum Addition/Subtraction for Perfect Square: Given non-square number $N$, find smallest integer $k$ to add/subtract so that $N \pm k$ becomes a perfect square. Use long division; if remainder $r \neq 0$, subtraction gives $N - r$. For addition, find next integer square $(q+1)^2 - N$.
SCERT UP Textbook Model: A gardener has $2025$ plants. He wants to plant them in such a way that the number of rows and columns are equal. Find the number of rows. Solution: $\sqrt{2025} = \sqrt{3^4 \cdot 5^2} = 3^2 \cdot 5 = 45$ rows.
Fractional Scaling Problems: If the area of a square field is given as a mixed fraction like $52\frac{1}{16} \text{ m}^2$, convert to improper fraction $\frac{833}{16}$ or use decimal conversion before applying square root algorithms.
UP Exam High-Yield Tip: In questions stating 'least number to be multiplied/divided to make a perfect square', always use Prime Factorization. In questions stating 'least number to be added/subtracted', always use Long Division.

4. Mathematical Edge Cases, Sign Conventions & Common Traps

Critical Distinction: The radical symbol $\sqrt{x}$ strictly represents the principal (positive) square root. Thus, $\sqrt{16} = +4$, NOT $\pm 4$. However, if $x^2 = 16$, then $x = \pm \sqrt{16} = \pm 4$. Distinguish carefully between evaluating a radical and solving a quadratic equation.
Edge Case of Zero and One: $\sqrt{0} = 0$, $\sqrt{1} = 1$. For numbers between $0$ and $1$ (e.g., $0.25$), the square root is greater than the number itself: $\sqrt{0.25} = 0.5 > 0.25$. This counter-intuitive property is a favorite trap in Super TET.
Common Calculation Trap: Confusing $\sqrt{a + b}$ with $\sqrt{a} + \sqrt{b}$. Note that $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. For example, $\sqrt{9 + 16} = \sqrt{25} = 5$, whereas $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$.
Trap Prevention Trick: When simplifying nested square roots like $\sqrt{7 + \sqrt{48}}$, express $48$ as $2 \cdot \sqrt{12} \cdot \sqrt{4}$ or recognize $(a+b)^2 = a^2+b^2+2ab$ to convert the inner expression into a binomial square $(2 + \sqrt{3})^2$.
Memory Mnemonic: ROOT — Radical is positive, Only pairs get out, Order pairs from right, Test decimals right.

Comprehensive Mathematical Matrix & Method Comparison

Parameter / DimensionMethod / Concept AMethod / Concept BExam Speed & Accuracy Takeaway
Best Suited ForPrime Factorization: Small to medium numbers with clear factors.Long Division: Large numbers, decimals, and non-perfect squares.Use Prime Factorization for 'multiply/divide' problems; Long Division for 'add/subtract'.
Handling DecimalsRequires conversion to fraction (numerator/denominator) first.Directly handles decimals by pairing from decimal point outward.Long Division is faster for decimals and avoids massive fraction factorizations.
Remainder AnalysisYields unpaired prime factors indicating deficient values.Yields exact numerical remainder $r$ for addition/subtraction problems.Long Division remainder directly gives the exact amount to subtract.
Time ComplexityFast for numbers $< 1000$; slow for $> 5$ digit numbers.Consistent step-by-step execution for any number of digits.Master Long Division to save crucial seconds in high-digit exam questions.
Practice Progress: 0 of 10 Answered
Score: 0 / 10
Question 1 of 10 Super TET Junior Mathematics

1. What is the least number by which $1470$ must be divided so that the quotient is a perfect square?

Correct Answer: Option A

Step 1: Perform prime factorization of $1470$. $1470 = 147 \times 10 = (3 \times 7^2) \times (2 \times 5) = 2^1 \times 3^1 \times 5^1 \times 7^2$. Step 2: Identify primes with odd exponents ($2^1, 3^1, 5^1$). Step 3: To make the quotient a perfect square, divide by the product of terms with odd exponents: $2 \times 3 \times 5 = 30$. Thus, dividing by $30$ leaves $7^2$, which is a perfect square.

Question 2 of 10 Super TET Junior Mathematics

2. Evaluate $\sqrt{0.0009} + \sqrt{0.04}$:

Correct Answer: Option A

Step 1: Evaluate the first term: $\sqrt{0.0009} = \sqrt{\frac{9}{10000}} = \frac{3}{100} = 0.03$. Step 2: Evaluate the second term: $\sqrt{0.04} = \sqrt{\frac{4}{100}} = \frac{2}{10} = 0.20$. Step 3: Add both values: $0.03 + 0.20 = 0.23$. Distractor $0.07$ arises from incorrect decimal placement ($\sqrt{0.04} = 0.02$). Higher accuracy ensures zero negative marking.

Question 3 of 10 Super TET Junior Mathematics

3. What is the greatest 4-digit number that is a perfect square?

Correct Answer: Option B

Step 1: The greatest 4-digit number is $9999$. Step 2: Apply long division to find $\sqrt{9999}$. Pairing gives $99\;99$. $9 \times 9 = 81$, remainder $18$. Bring down $99$ to get $1899$. Trial divisor $18y \times y \le 1899 \implies 189 \times 9 = 1701$. Remainder is $198$. Step 3: Subtract remainder from $9999$: $9999 - 198 = 9801$. Step 4: Verify $99^2 = 9801$. Option $9801$ is correct.

Question 4 of 10 Super TET Junior Mathematics

4. Find the square root of $11 + 2\sqrt{30}$:

Correct Answer: Option A

Step 1: Use the surd identity $\sqrt{a + b + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b}$. Step 2: Here, $a + b = 11$ and $ab = 30$. Step 3: Find two numbers whose sum is $11$ and product is $30$; these are $6$ and $5$. Step 4: Therefore, $\sqrt{11 + 2\sqrt{30}} = \sqrt{(\sqrt{6} + \sqrt{5})^2} = \sqrt{6} + \sqrt{5}$. Option 0 is correct.

Question 5 of 10 Super TET Junior Mathematics

5. What least number must be added to $5607$ to make the sum a perfect square?

Correct Answer: Option A

Step 1: Apply long division on $56\;07$. $7 \times 7 = 49$, remainder $7$. Bring down $07$ to get $707$. Trial divisor $14y \times y \le 707 \implies 144 \times 4 = 576$. Remainder is $707 - 576 = 131$. Step 2: Since $74^2 = 5476 < 5607$, the next integer is $75$. Step 3: Calculate $75^2 = 5625$. Step 4: Number to be added = $5625 - 5607 = 18$. Option 0 is correct.

Question 6 of 10 Super TET Junior Mathematics

6. If $\sqrt{1 + \frac{x}{144}} = \frac{13}{12}$, what is the value of $x$?

Correct Answer: Option A

Step 1: Square both sides of the equation: $1 + \frac{x}{144} = \left(\frac{13}{12}\right)^2 = \frac{169}{144}$. Step 2: Simplify: $\frac{x}{144} = \frac{169}{144} - 1 = \frac{169 - 144}{144} = \frac{25}{144}$. Step 3: Equating numerators gives $x = 25$. Option 0 is correct.

Question 7 of 10 Super TET Junior Mathematics

7. Evaluate $\sqrt{24} + \sqrt{216} - \sqrt{96}$:

Correct Answer: Option B

Step 1: Simplify each surd by extracting square factors. $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$. Step 2: $\sqrt{216} = \sqrt{36 \times 6} = 6\sqrt{6}$. Step 3: $\sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}$. Step 4: Combine terms: $2\sqrt{6} + 6\sqrt{6} - 4\sqrt{6} = (2 + 6 - 4)\sqrt{6} = 4\sqrt{6}$. Option 1 is correct.

Question 8 of 10 Super TET Junior Mathematics

8. A troop of soldiers can be arranged in rows of $12, 15, 18,$ and $20$ soldiers such that each row has an equal number, and they form a solid square. What is the minimum number of soldiers in the troop?

Correct Answer: Option A

Step 1: Find the LCM of $12, 15, 18,$ and $20$. Prime factorizations: $12=2^2\cdot3, 15=3\cdot5, 18=2\cdot3^2, 20=2^2\cdot5$. $\text{LCM} = 2^2 \cdot 3^2 \cdot 5 = 4 \times 9 \times 5 = 180$. Step 2: Prime factors of $180$ are $2^2 \cdot 3^2 \cdot 5^1$. To make it a square, exponent of $5$ must be even, so multiply by $5$. Step 3: Minimum soldiers = $180 \times 5 = 900$. Option 0 is correct.

Question 9 of 10 Super TET Junior Mathematics

9. What is the square root of $0.9$ up to two decimal places?

Correct Answer: Option A

Step 1: Set up long division for $0.90\;00\;00$. Step 2: First digit after decimal: $9 \times 9 = 81$, remainder $9$. Bring down $00$ to get $90$. Step 3: Double quotient ($0.9 \times 2 = 1.8$, trial divisor $184 \times 4 = 736$). Remainder $164$. Bring down $00$ to get $16400$. Step 4: Trial divisor $1885 \times 5 = 9425$. Thus, $\sqrt{0.9} \approx 0.948$, which rounds to $0.95$. Option 0 is correct.

Question 10 of 10 Super TET Junior Mathematics

10. If $x = \sqrt{5 + \sqrt{5 + \sqrt{5 + \dots \infty}}},$ then which of the following inequalities holds true for $x$?

Correct Answer: Option A

Step 1: Notice that $x = \sqrt{5 + x}$. Step 2: Square both sides: $x^2 = 5 + x \implies x^2 - x - 5 = 0$. Step 3: Using the quadratic formula, $x = \frac{1 \pm \sqrt{1 + 20}}{2} = \frac{1 + \sqrt{21}}{2}$. Step 4: Since $\sqrt{21}$ lies between $4$ and $5$ (specifically $\approx 4.58$), $x = \frac{1 + 4.58}{2} = 2.79$. Thus, $2 < x < 3$. Option 0 is correct.

5-Minute Exam Simulation Mode

Experience the real pressure of Super TET Junior (Part 2C Mathematics). You will face 10 randomized speed-math questions with negative marking (-1 for incorrect answers) and a strict 5-minute countdown.

Time Limit:

5 Minutes (300s)

Marking Scheme:

+3 Marks | -1 Negative

Passing Benchmark:

21+ Marks (70%)

High-Yield Rapid Recall Formulas & Laws (Must-Remember Pointers)

#1
Principal square root $\sqrt{x}$ is always non-negative for any real $x \ge 0$.
#2
Units digit of a perfect square can only be $0, 1, 4, 5, 6,$ or $9$.
#3
Prime factorization requires all prime power exponents to be even for a perfect square.
#4
Long division method pairs digits from decimal point leftwards for integers and rightwards for decimals.
#5
$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$ are fundamental radical axioms.
#6
For numbers between $0$ and $1$, the square root is strictly greater than the number itself.
#7
To find the least number to add/subtract to make a square, always use the Long Division remainder method.
#8
To find the least number to multiply/divide to make a square, always use the Prime Factorization method.

Speed Math & Memory Mnemonics Vault (Exam Retention Anchors)

Square Root Method Selector

Memory Key: PAMS

P - Prime factors for multiply/divide, A - Addition/subtraction uses long division, M - Match digit pairs, S - Simplify radicals first.

Long Division Steps

Memory Key: PADD

P - Pair digits, A - Appraise root, D - Double divisor, D - Drop next pair.

Negative Marking Traps & Common Blunders (-1 Penalty Avoidance)

Negative Marking Alert #1:

Negative Marking Trap 1: Assuming $\sqrt{x^2} = x$ universally without considering absolute values for negative domains (i.e., $\sqrt{x^2} = |x|$).

Negative Marking Alert #2:

Negative Marking Trap 2: Confusing the least number to be multiplied with the least number to be added; mixing up prime factorization and long division algorithms.

Negative Marking Alert #3:

Negative Marking Trap 3: Incorrect decimal pairing in long division (e.g., pairing $0.9$ as $0.9$ instead of $0.90$), leading to severe calculation errors.

Negative Marking Alert #4:

Negative Marking Trap 4: Assuming $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$, violating basic radical algebraic rules.

Self-Assessment Checklist

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