Question

Can we somehow simplify the expression (6^21)(10^7)/30^7 and express it as a product of primes, each raised to an integer power?

a) (2^7)(3^7)(5^7)
b) (2^7)(3^7)(5^6)
c) (2^6)(3^7)(5^7)
d) (2^7)(3^6)(5^7)

Answer 1

To simplify the expression (6^21)(10^7)/30^7 and express it as a product of primes, we need to prime factorize each number and then cancel out common factors. The simplified expression is (2^7)(3^7)(5^7).

Step-by-step explanation:

To simplify the expression (6^21)(10^7)/30^7 and express it as a product of primes, we need to rewrite the numbers in terms of their prime factorization. Let's break it down step-by-step:

1. Prime factorize 6: 6 can be expressed as 2 * 3.
2. Prime factorize 10: 10 can be expressed as 2 * 5.
3. Prime factorize 30: 30 can be expressed as 2 * 3 * 5.
4. Now, let's rewrite the expression using prime factorization:
5. (6^21)(10^7)/30^7 = (2^21 * 3^21 * 5^21)(2^7 * 5^7)/(2^7 * 3^7 * 5^7)
6. Cancel out the common factors in the numerator and denominator:
7. (2^21 * 3^21 * 5^21)(2^7 * 5^7)/(2^7 * 3^7 * 5^7) = (2^({21+7}-7) * 3^{21-7} * 5^{21-7}) = (2^21 * 3^14 * 5^14)
8. Write the expression as a product of primes, each raised to an integer power:
9. (2^21 * 3^14 * 5^14) = (2^7 * 2^7 * 2^7 * 2) * (3^7 * 3^7) * (5^7 * 5^7) = (2^7)(3^7)(5^7)
10. Therefore, the simplified expression is (2^7)(3^7)(5^7) which corresponds to option a.