Express 1728 and 2025 in terms of their prime factors. Hence evaluate:
((1728)^1/3))/sq(2025)
1728 = 12^3
2025 = 45^2
You can reduce those to prime factors if you want, but it's pretty clear what's the answer
Express 1728 and 2025 in terms of their prime factors
To express 1728 and 2025 in terms of their prime factors, we can factorize them as follows:
1728 = 2^6 * 3^3
2025 = 3^4 * 5^2
Now, let's evaluate ((1728)^(1/3))/sqrt(2025) step by step.
First, let's simplify the numerator, (1728)^(1/3):
(1728)^(1/3) = (2^6 * 3^3)^(1/3)
= 2^2 * 3
Next, let's simplify the denominator, sqrt(2025):
sqrt(2025) = sqrt(3^4 * 5^2)
= 3^2 * 5
= 9 * 5
= 45
Finally, let's evaluate the expression:
((1728)^(1/3))/sqrt(2025) = (2^2 * 3) / 45
Therefore, the simplified expression is (2^2 * 3) / 45.
To express a number in terms of its prime factors, we need to find the prime numbers that multiply together to give us the original number.
1. Prime factorization of 1728:
We start by dividing the number by the smallest prime number, which is 2. Since 1728 is an even number, it divides evenly by 2. We do this repeatedly until it is no longer divisible by 2:
1728 ÷ 2 = 864
864 ÷ 2 = 432
432 ÷ 2 = 216
216 ÷ 2 = 108
108 ÷ 2 = 54
54 ÷ 2 = 27
Now, since 27 is an odd number, we move on to the next prime number, which is 3. We divide 27 by 3:
27 ÷ 3 = 9
We continue with the next prime number, which is also 3:
9 ÷ 3 = 3
Now we have reached a prime number, which is 3. So the prime factorization of 1728 is:
1728 = 2^6 × 3^3
2. Prime factorization of 2025:
We follow the same process as above. Since 2025 ends in 5, it means it is divisible by 5:
2025 ÷ 5 = 405
We can further divide 405 by 3:
405 ÷ 3 = 135
Again, we divide 135 by 3:
135 ÷ 3 = 45
45 is divisible by 3 as well:
45 ÷ 3 = 15
Finally, we have 15, which is divisible by 5:
15 ÷ 5 = 3
Now we have reached a prime number, which is 3. So the prime factorization of 2025 is:
2025 = 3^4 × 5^2
Now we can evaluate the expression:
((1728)^(1/3)) / √(2025)
We can simplify this by substituting the prime factorizations of both numbers:
((2^6 × 3^3)^(1/3)) / √(3^4 × 5^2)
Using the exponent properties, raising a number to the power of 1/3 is equivalent to taking the cube root of that number. Simplifying further:
(2^6)^(1/3) × (3^3)^(1/3) / √(3^4 × 5^2)
Now we can merge the exponents inside the parentheses:
2^(6/3) × 3^(3/3) / √(3^4 × 5^2)
Simplifying the exponents:
2^2 × 3 / √(3^4 × 5^2)
Calculating the values:
4 × 3 / √(81 × 25)
Multiplying and simplifying:
12 / √(2025)
We already found that the prime factorization of 2025 is 3^4 × 5^2, so:
12 / √(3^4 × 5^2)
Now, we can simplify √(3^4 × 5^2) as follows:
√(3^4 × 5^2) = √(3^2 × 3^2 × 5^2)
= √(3^2) × √(3^2) × √(5^2)
= 3 × 3 × 5
= 45
Substituting this value back into the expression:
12 / 45
However, we can simplify this fraction further by dividing both the numerator and denominator by their greatest common divisor, which is 3:
12 ÷ 3 / 45 ÷ 3
4 / 15
Therefore, the value of the expression is 4/15.