Real Numbers — Telangana (SSC) Class 9 Mathematics Solutions (Free)

A rational number is any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0 — this includes whole numbers, fractions, and decimals that either terminate or repeat in a fixed pattern. An irrational number cannot be written this way — its decimal expansion goes on forever without ever settling into a repeating pattern. √16 = 4, an integer, which can be written as 4/1 → RATIONAL √17 — 17 is not a perfect square, and the square root of any non-perfect-square whole number is always irrational; its decimal expansion (≈4.1231...) never terminates or repeats → IRRATIONAL π ≈ 3.14159265... — its decimal expansion has been calculated to trillions of digits and never repeats; this has been mathematically proven, not just observed → IRRATIONAL 22/7 — despite being a common approximation FOR π, 22/7 is itself an exact ratio of two integers (22 and 7), so it is RATIONAL. (It is only an approximation to the irrational π, not equal to it.) 0.333... — a repeating decimal is always rational, since it can always be converted to a fraction: 0.333... = 1/3 → RATIONAL √(4/9) = 2/3, a ratio of two integers → RATIONAL Final classification: Rational — √16, 22/7, 0.333..., √(4/9); Irrational — √17, π.

The key technique is to write each square root as (perfect square) × (remaining factor), so that a whole number can be pulled outside the root, leaving matching radicals that can then be combined like terms. √72 = √(36 × 2) = 6√2 (since 36 is the largest perfect square factor of 72) √50 = √(25 × 2) = 5√2 (since 25 is the largest perfect square factor of 50) √8 = √(4 × 2) = 2√2 (since 4 is the largest perfect square factor of 8) Now combine, since all three terms share the same radical part (√2): 6√2 − 5√2 + 2√2 = (6 − 5 + 2)√2 = 3√2 The simplified answer is 3√2.

A denominator containing a sum or difference of square roots is rationalised by multiplying both the numerator and denominator by the CONJUGATE of the denominator — the same two terms with the sign between them flipped. This works because (a + b)(a − b) = a² − b², which eliminates the square roots entirely. The conjugate of (√5 + √3) is (√5 − √3). 1/(√5 + √3) × (√5 − √3)/(√5 − √3) = (√5 − √3) / [(√5)² − (√3)²] = (√5 − √3) / (5 − 3) = (√5 − √3) / 2 The rationalised form is (√5 − √3)/2 — a denominator with no square root remaining.

Two laws of exponents are needed here: - Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents when multiplying same base) - Quotient rule: aᵐ/aⁿ = aᵐ⁻ⁿ (subtract exponents when dividing same base) Step 1: Apply the product rule to the numerator 2³ × 2⁴ = 2³⁺⁴ = 2⁷ Step 2: Apply the quotient rule 2⁷/2⁵ = 2⁷⁻⁵ = 2² Step 3: Evaluate 2² = 4 The simplified value is 4.

This is proved by contradiction — assuming the opposite of what we want to prove, then showing that assumption leads to something impossible. Step 1: Assume √2 IS rational. Then it can be written as √2 = p/q, where p and q are integers with no common factor (the fraction is in its lowest terms), and q ≠ 0. Step 2: Square both sides: 2 = p²/q², so p² = 2q². Step 3: Since p² = 2q², p² is an even number (2 times something). If p² is even, p itself must be even (an odd number squared always gives an odd number, so only an even p can give an even p²). Let p = 2k for some integer k. Step 4: Substitute back: (2k)² = 2q², so 4k² = 2q², so q² = 2k². Step 5: By the same logic as Step 3, q² being even means q must be even too. Step 6: But now BOTH p and q are even, meaning they share a common factor of 2 — this directly contradicts our starting assumption in Step 1 that p and q have NO common factor. Step 7: Since assuming √2 is rational leads to a contradiction, the assumption must be false. Therefore, √2 is irrational.

Three laws of exponents apply here in sequence: - Power of a power: (aᵐ)ⁿ = aᵐⁿ - Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ - Quotient rule: aᵐ/aⁿ = aᵐ⁻ⁿ Step 1: Simplify the power of a power (3²)³ = 3²ˣ³ = 3⁶ Step 2: Multiply by 3⁻⁵ using the product rule 3⁶ × 3⁻⁵ = 3⁶⁺⁽⁻⁵⁾ = 3¹ = 3 Step 3: Divide by 3⁴ using the quotient rule 3¹/3⁴ = 3¹⁻⁴ = 3⁻³ Step 4: Convert the negative exponent to a fraction 3⁻³ = 1/3³ = 1/27 The simplified value is 1/27.

Scientific notation writes any number as (a number between 1 and 10) × 10ⁿ. Step 1: Move the decimal point right until only one non-zero digit remains before it — from 0.000047, moving the decimal point 5 places to the right gives 4.7. Step 2: Since we moved the decimal RIGHT (making the number appear larger), the exponent must be NEGATIVE, to compensate and bring the value back down: 0.000047 = 4.7 × 10⁻⁵ Verification: 4.7 × 10⁻⁵ = 4.7 ÷ 100,000 = 0.000047 ✓ The scientific notation is 4.7 × 10⁻⁵.

When both sides of an equation can be expressed as powers of the SAME base, the exponents themselves must be equal. Step 1: Express 8 as a power of 2 8 = 2 × 2 × 2 = 2³ Step 2: Now the equation reads 2ˣ = 2³ Step 3: Since the bases are equal (both are base 2), the exponents must be equal x = 3

As before, break each term into (perfect square) × (remaining factor): √50 = √(25 × 2) = 5√2 √18 = √(9 × 2) = 3√2 √8 = √(4 × 2) = 2√2 Combine, since all three share the radical √2: 5√2 + 3√2 − 2√2 = (5 + 3 − 2)√2 = 6√2 The simplified answer is 6√2.

Step 1: Express 27 as a power of 3 27 = 3 × 3 × 3 = 3³ Step 2: Now the equation reads 3^(2x − 1) = 3³ Step 3: Since the bases are equal, the exponents must be equal 2x − 1 = 3 Step 4: Solve for x 2x = 4 x = 2 Verification: 3^(2(2) − 1) = 3^(4 − 1) = 3³ = 27 ✓