Prime Factorisation

Step 1: The prime factorisation of 72 is (2^3\times3^2). Step 2: Comparing with the given form, the power of 2 is (a). Step 3: Therefore, (a=3).

Step 1: The prime factorisation of 90 is (2\times3^2\times5). Step 2: In the given form, the power of 3 is (b). Step 3: Comparing gives (b=2).

Step 1: Calculate (2^2=4). Step 2: (4\times3\times5=60). Step 3: To get the number from prime factorisation, multiply all factors.

Step 1: Calculate (2^3=8) and (3^2=9). Step 2: (8\times9=72). Step 3: Evaluating powers first gives the answer quickly.

Step 1: (2^3=8). Step 2: (8\times5=40). Step 3: Multiply the factors to convert prime factorisation into the number.

The governing concept is evaluation of a numerical expression containing a power and multiplication. In m = 3^2 × 7, the exponent 2 means that 3 is multiplied by itself: 3^2 = 3 × 3 = 9. After evaluating the power, multiply the result by 7: m = 9 × 7 = 63. Therefore, option A is correct. Option B, 42, does not result from correctly evaluating the square. Options C and D also fail because neither equals 9 × 7. The answer can be checked through prime factorisation: 63 = 3 × 3 × 7 = 3^2 × 7, exactly matching the given expression. Thus the numerical value and its factorised form are consistent.

Step 1: In prime factorisation, every factor must be prime. Step 2: 2, 3, and 5 are prime. Step 3: 4, 6, and 10 are composite, so they cannot remain in the final prime form.

Step 1: A final prime factorisation must not contain a composite number. Step 2: 4 is composite, so (4\times3\times5) is not final form. Step 3: Change 4 into (2^2).

Step 1: Divide 64 repeatedly by 2. Step 2: Six factors of 2 give (64=2^6). Step 3: 4 and 8 are composite, so do not write them in final prime form.

Step 1: Divide 81 repeatedly by 3. Step 2: Four factors of 3 give (81=3^4). Step 3: 9 and 27 are composite, so write the power of 3 in final prime form.

Step 1: 121 is a square number. Step 2: (121=11\times11=11^2). Step 3: Since 11 is prime, (11^2) is the correct prime factorisation.

Step 1: 49 can be written as (7\times7). Step 2: Since 7 is prime, (49=7^2). Step 3: In a square number, the same prime may appear twice.

Step 1: Write 27 as (3\times9). Step 2: Since (9=3^2), (27=3^3). Step 3: 9 is composite, so the final form is (3^3).

Step 1: Divide 32 repeatedly by 2. Step 2: Five factors of 2 give (32=2^5). Step 3: 4 and 8 are composite, so do not write them in final form.

Step 1: 16 is (2\times2\times2\times2). Step 2: Therefore, (16=2^4). Step 3: 4 and 8 are composite, so write the power of 2 in final prime form.

Step 1: 25 is (5\times5). Step 2: Since 5 is prime, (25=5^2). Step 3: If the same prime appears twice, write power 2.

Step 1: Write (80=16\times5). Step 2: Since (16=2^4), (80=2^4\times5). Step 3: 16 is composite, so change it into (2^4).

Step 1: Write (98=2\times49). Step 2: Since (49=7^2), (98=2\times7^2). Step 3: 49 is composite, so write (7^2) in the final form.

Step 1: Write (105=15\times7). Step 2: (15=3\times5), so (105=3\times5\times7). Step 3: 15, 21, and 35 are composite, so do not keep them in the final answer.

Step 1: Calculate (2^2=4) and (3^2=9). Step 2: (4\times9=36). Step 3: Multiply to get the number from prime factorisation.

Prime factorisation lists the prime factors with their powers. Compute the power first: \(5^2=25\). Then multiply by 2: \(2\times25=50\). So the number is 50. Why other options are wrong: 25 lacks the factor 2; 100 equals \(2^2\times5^2\) (an extra factor 2); 40 equals \(2^3\times5\). Exam tip: evaluate powers first or multiply the listed prime factors directly to get the number.

Step 1: The prime factors are given. Step 2: (2\times3\times7=42). Step 3: When no power is written, each prime is taken once.

Answer: A, 75. Prime factorisation means expressing a number as a product of prime numbers, with powers showing repeated factors. First evaluate the power: 5^2 = 5 × 5 = 25. Then multiply by 3: 3 × 25 = 75. Therefore, the required number is 75. Option A is correct because 75 = 3 × 5 × 5 = 3 × 5^2. Option B, 45, has factorisation 3^2 × 5, so the exponent of 3 is wrong and only one 5 is present. Option C, 90, equals 2 × 3^2 × 5 and contains the extra prime factor 2. Option D, 150, equals 2 × 3 × 5^2 and also contains an unwanted factor 2. Memory cue: calculate powers first, then multiply; do not confuse the exponent with multiplication by the exponent.

Step 1: Calculate (2^4=16). Step 2: (16\times3=48). Step 3: It is easier to find the value of prime powers first.

Step 1: A final prime factorisation contains only prime numbers. Step 2: A composite factor must be broken further. Step 3: In exams, do not leave factors like 6, 8, or 10 at the end.