Decimal expansion of rational numbers

Step 1: The reduced denominator contains only powers of (2) and (5). Step 2: The number of decimal places equals the larger exponent. Here the larger exponent is (7). Step 3: Do not add the exponents; use the larger one.

Step 1: (154=2\cdot 7\cdot 11). Step 2: After cancelling (2\cdot 7\cdot 11), the denominator becomes (5^3). So the decimal terminates after (3) places. Step 3: Extra factors may cancel with the numerator, so reduce first.

Step 1: For a terminating decimal, the reduced denominator must contain only (2) and (5). Step 2: The factors (3^3) and (11) must be cancelled, so the least (n) is (3^3\cdot 11=297). Step 3: For the smallest value, cancel only the unwanted prime factors.

Step 1: Let (x=0.1242424\ldots). Step 2: (10x=1.242424\ldots) and (1000x=124.242424\ldots), so (990x=123) and (x=\frac{123}{990}=\frac{41}{330}). Step 3: First identify the lengths of the non-repeating and repeating parts.

Step 1: (42=2\cdot 3\cdot 7). Step 2: After cancellation, the denominator becomes (2\cdot 5^4). The larger exponent is (4), so the decimal terminates after (4) places. Step 3: Check only the remaining denominator after cancellation.

Step 1: The denominator has only powers of (2) and (5), so the decimal terminates. Step 2: The number of decimal places equals the larger of (a) and (b). For exactly (8) places, (\max(a,b)=8). Step 3: Remember the larger exponent in such questions.

Step 1: Look for any factor other than (2) and (5) that remains in the denominator. Step 2: In (\frac{50}{2\cdot 5^2\cdot 7}), (50=2\cdot 5^2) cancels, but (7) remains. So the decimal is non-terminating recurring. Step 3: The remaining prime factors after cancellation decide the type.

Step 1: (0.00064=\frac{64}{100000}). Step 2: Reducing by the greatest common factor (32) gives (\frac{2}{3125}). So the denominator is (3125). Step 3: Reduce carefully; repeated division by (2) is safe here.

Step 1: The fraction is in lowest form, so the factor (17) will not cancel. Step 2: The reduced denominator has (17) besides (2) and (5). Therefore the decimal is non-terminating recurring. Step 3: A non-terminating decimal of a rational number is recurring.

Step 1: (10^8=2^8\cdot 5^8). Step 2: The given denominator is (2^5\cdot 5^8), so it lacks (2^3). Step 3: To form (10^k), increase the prime factor with the smaller exponent.

Step 1: (0.\overline{36}=\frac{36}{99}). Step 2: (\frac{36}{99}=\frac{4}{11}), so the reduced denominator is (11). Step 3: For a purely recurring decimal, first use a denominator of (9)'s and then reduce.

Step 1: (0.\overline{018}) has a repeating block, so it is rational. Step 2: It does not end, so it is not a terminating decimal. The other options are either terminating or irrational. Step 3: Recurring decimals are rational.

Step 1: (225=3^2\cdot 5^2). Step 2: After cancellation, the denominator becomes (2^4\cdot 5^3). The larger exponent is (4), so the decimal terminates after (4) places. Step 3: Powers present in the numerator can reduce the decimal length.

Step 1: (540=2^2\cdot 3^3\cdot 5). Step 2: For a terminating decimal, (3^3) must cancel completely from the denominator. So (x) must contain (27). Step 3: (2) and (5) may remain, but (3) must not.

Step 1: The decimal does not terminate. Step 2: The number of (2)'s keeps changing, so no fixed repeating block is formed. Hence it is non-terminating non-recurring. Step 3: Repeated-looking digits are not enough; a fixed recurring block is needed.

Step 1: The denominator has only powers of (2) and (5). Step 2: Since (a>b), the larger exponent is (a). For exactly (6) decimal places, (a=6). Step 3: When a comparison is given, identify the larger exponent immediately.

Step 1: We need (10^5=2^5\cdot 5^5). Step 2: The denominator (2^3\cdot 5^5) lacks (2^2). Multiplying numerator and denominator by (4) gives (N=7\cdot 4=28). Step 3: Multiply by the missing part to make the denominator (10^k).

Step 1: (64000=2^9\cdot 5^3), so it would give (9) places, not (6). Step 2: (15625=5^6), so (\frac{9}{15625}) terminates exactly after (6) places. Step 3: Calculate prime powers carefully.

Step 1: (15625=5^6). Step 2: The powers are (0) for (2) and (6) for (5). So the larger exponent is (6). Step 3: Practise identifying higher powers of (5).

Step 1: Dividing (\frac{35}{280}) by (35) gives (\frac{1}{8}). Step 2: Since (8=2^3), the decimal terminates. The reason correctly explains the assertion. Step 3: In assertion-reason questions, also check whether the reason explains the assertion.