Prime Factorisation

Step 1: The prime factorisation of 600 is (2^3\times3\times5^2). Step 2: In the given form, the power of 2 is (a). Step 3: Comparing gives (a=3).

Step 1: (675=27\times25). Step 2: (27=3^3) and (25=5^2), so (675=3^3\times5^2). Step 3: Comparing gives (a=3).

Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5=720). Step 3: Evaluating powers first makes calculation easier.

This question applies the meaning of an exponent in a prime factorisation. In 2 × 3^4, calculate the power first: 3^4 = 3 × 3 × 3 × 3 = 81. Then multiply by the factor 2: 2 × 81 = 162. Therefore option A, 162, is correct. Option B is only 3^4 and omits the factor 2. Option C is twice the correct result, and option D includes an incorrect extra factor. A factorisation check confirms the result: 162 = 2 × 81 = 2 × 3^4. This illustrates that a prime factorisation records both the prime factors and their multiplicities, represented here by the exponent 4 on 3.

Step 1: (2^3=8). Step 2: (8\times3\times7=168). Step 3: Multiply to get the number from prime factorisation.

The governing idea is evaluating each power in a factorised numerical expression and then multiplying the results. Here, 3^2 = 3 × 3 = 9 and 5^2 = 5 × 5 = 25. Therefore m = 9 × 25 = 225. Option A is correct. Another efficient check is to group equal exponents: 3^2 × 5^2 = (3 × 5)^2 = 15^2 = 225, using the law a^n b^n = (ab)^n. Option B is too small and reflects an incorrect calculation, while 300 and 450 do not follow from the stated factors. The prime factorisation of 225 is 3^2 × 5^2, confirming the answer independently.

Step 1: In final prime factorisation, every factor must be prime. Step 2: 2, 3, 5, 7, and 11 are prime. Step 3: 6, 21, 35, and 33 are composite, so they cannot remain in final form.

Step 1: Final prime factorisation must not contain a composite factor. Step 2: 10 is composite, so (10\times3^2\times7) is not final form. Step 3: Change 10 into (2\times5).

Step 1: Divide 2048 repeatedly by 2. Step 2: Eleven factors of 2 give (2048=2^{11}). Step 3: 4, 32, and 64 are composite, so write the power of 2 in final form.

Step 1: Divide 729 repeatedly by 3. Step 2: Six factors of 3 give (729=3^6). Step 3: 9 and 27 are composite, so keep prime base 3.

Step 1: (361=19\times19). Step 2: Since 19 is prime, (361=19^2). Step 3: In a square number, the same prime appears twice.

Step 1: Write (1331=11\times121). Step 2: (121=11^2), so (1331=11^3). Step 3: 121 is composite, so write (11^3) in the final form.

Step 1: Divide 4096 repeatedly by 2. Step 2: Twelve factors of 2 give (4096=2^{12}). Step 3: 64 and 16 are composite, so they are not final prime forms.

Step 1: Write (3125=5\times625). Step 2: Since (625=5^4), (3125=5^5). Step 3: 25 and 125 are composite forms, so write a power of 5 in the final answer.

Step 1: Recognise (2744=14^3). Step 2: Since (14=2\times7), (2744=2^3\times7^3). Step 3: 14 is composite, so write prime bases in the final form.

Step 1: Write (1536=512\times3). Step 2: (512=2^9), so (1536=2^9\times3). Step 3: Do not keep 512 in the final form; write (2^9).

Step 1: Write (1080=108\times10). Step 2: (108=2^2\times3^3) and (10=2\times5), so (1080=2^3\times3^3\times5). Step 3: Count powers of 2 and 3 separately.

Step 1: Write (1470=30\times49). Step 2: (30=2\times3\times5) and (49=7^2), so (1470=2\times3\times5\times7^2). Step 3: Convert 49 into (7^2).

Step 1: Write (1980=198\times10). Step 2: (198=2\times3^2\times11) and (10=2\times5), so (1980=2^2\times3^2\times5\times11). Step 3: Since 2 appears twice, write (2^2).

Step 1: Calculate (2^3=8) and (7^2=49). Step 2: (8\times3\times49=1176). Step 3: Evaluating powers first makes calculation easier.

Step 1: (5^2=25) and (7^2=49). Step 2: (25\times49=1225). Step 3: Both powers are 2 in this square form, so multiply carefully.

Step 1: Multiply all given prime factors. Step 2: (2\times3\times5\times7\times11=2310). Step 3: When no power is written, each prime is taken once.

Step 1: Calculate (2^3=8), (3^2=9), and (5^2=25). Step 2: (8\times9\times25=1800). Step 3: Simplify powers first, then multiply.

Step 1: Calculate (2^4=16) and (3^4=81). Step 2: (16\times81=1296). Step 3: Evaluate prime powers and then multiply.

Step 1: When a prime repeats, powers are used. Step 2: For example, (2\times2\times2) is written as (2^3). Step 3: This makes the answer short, clear, and easy to read in exams.