Use prime factors of 1764 and 2744 to evaluate √1764/³√2744
1764 = 2^2 3^2 7^2
2744 = 2^3 7^3
so √1764 / ∛2744 = (2*3*7)/(2*7) = 3
To evaluate √1764/³√2744 using prime factors, we first need to find the prime factorization of the numbers 1764 and 2744.
To find the prime factorization of a number, you need to break it down into its prime factors, which are the prime numbers that divide the given number exactly.
Let's find the prime factorization of 1764:
1. Start by dividing the number by the smallest prime number, which is 2. 1764 ÷ 2 = 882.
2. Continue dividing by 2 until you can no longer divide evenly: 882 ÷ 2 = 441. 441 is an odd number, so we move to the next prime number.
3. Divide by the next prime number, 3. 441 ÷ 3 = 147.
4. Continue dividing by 3 until you can no longer divide evenly: 147 ÷ 3 = 49. We move to the next prime number since 49 is a perfect square.
5. Divide by the next prime number, 7. 49 ÷ 7 = 7.
6. Divide by 7 one more time: 7 ÷ 7 = 1.
The prime factorization of 1764 is 2² × 3² × 7² = 2 × 2 × 3 × 3 × 7 × 7 = 2² × 3² × 7².
Now let's find the prime factorization of 2744:
1. Divide by 2: 2744 ÷ 2 = 1372.
2. Divide by 2 again: 1372 ÷ 2 = 686.
3. Divide by 2 once more: 686 ÷ 2 = 343.
4. Divide by 7: 343 ÷ 7 = 49.
5. Divide by 7 once more: 49 ÷ 7 = 7.
6. Divide by 7 one more time: 7 ÷ 7 = 1.
The prime factorization of 2744 is 2³ × 7³ = 2 × 2 × 2 × 7 × 7 × 7 = 2³ × 7³.
Now, let's use the prime factorization to evaluate √1764/³√2744:
√1764 = √(2² × 3² × 7²) = √(2²) × √(3²) × √(7²) = 2 × 3 × 7 = 42.
³√2744 = ³√(2³ × 7³) = ³√(2³) × ³√(7³) = 2 × 7 = 14.
Therefore, √1764/³√2744 = 42/14 = 3.
To evaluate √1764/³√2744 using the prime factors of 1764 and 2744, let's begin by finding the prime factorization of both numbers.
Prime factorization of 1764:
1764 = 2^2 × 3^2 × 7^2
Prime factorization of 2744:
2744 = 2^3 × 7^3
Now, let's simplify the expression:
√1764/³√2744
Taking the square root of 1764:
√1764 = √(2^2 × 3^2 × 7^2)
= 2 × 3 × 7
= 42
Taking the cube root of 2744:
³√2744 = ³√(2^3 × 7^3)
= 2 × 7
= 14
So, √1764/³√2744 simplifies to 42/14.
Further simplifying, we can cancel out the common factor of 14:
42/14 = 3
Therefore, √1764/³√2744 equals 3.