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Given that Z=2^(2)×3^(2)×5^(2)×7^(p), express 70 Z as a product of powers of its prime factors.

Given that $Z=2^{2} \times 3^{2} \times 5^{2} \times 7^{p}$ , express $70 Z$ as a product of powers of its prime factors.

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Prime factorization with exponents

Full solution

Q. Given that

Z=2^{2} \times 3^{2} \times 5^{2} \times 7^{p}

, express

70 Z

as a product of powers of its prime factors.

Identify Prime Factors: Identify the prime factors of $70$ . The prime factors of $70$ are $2$ , $5$ , and $7$ .
Express as Product: Express $70$ as a product of powers of its prime factors. Since $70 = 2 \times 5 \times 7$ , we can write it as $2^{1}\times5^{1}\times7^{1}$ .
Combine with Z: Combine the prime factorization of $70$ with $Z$ . We have $Z = 2^{2}\times3^{2}\times5^{2}\times7^{p}$ and $70 = 2^{1}\times5^{1}\times7^{1}$ . Multiplying these together, we get $70Z = (2^{2}\times3^{2}\times5^{2}\times7^{p}) \times (2^{1}\times5^{1}\times7^{1})$ .
Combine Like Terms: Use the property of exponents that states $a^{m} \times a^{n} = a^{m+n}$ to combine like terms. For the prime factor $2$ , we have $2^{2} \times 2^{1} = 2^{2+1} = 2^{3}$ . For the prime factor $5$ , we have $5^{2} \times 5^{1} = 5^{2+1} = 5^{3}$ . For the prime factor $7$ , we have $7^{p} \times 7^{1} = 7^{p+1}$ . The prime factor $3$ remains unchanged as it is not a factor of $70$ .
Final Expression: Write the final expression for $70Z$ as a product of powers of its prime factors. The final expression is $70Z = 2^{3}\times3^{2}\times5^{3}\times7^{p+1}.$

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