Given that Z=2^(2)×3^(2)×5^(2)×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 × 5 × 7, we can write it as 2^(1)×5^(1)×7^(1).Combine with Z: Combine the prime factorization of 70 with Z. We have Z = 2^(2)×3^(2)×5^(2)×7^(p) and 70 = 2^(1)×5^(1)×7^(1). Multiplying these together, we get 70Z = (2^(2)×3^(2)×5^(2)×7^(p)) × (2^(1)×5^(1)×7^(1)).Combine Like Terms: Use the property of exponents that states a^(m) × a^(n) = a^(m+n) to combine like terms. For the prime factor 2, we have 2^(2) × 2^(1) = 2^(2+1) = 2^(3). For the prime factor 5, we have 5^(2) × 5^(1) = 5^(2+1) = 5^(3). For the prime factor 7, we have 7^(p) × 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)×3^(2)×5^(3)×7^(p+1).

Home

Math Problems

Grade 6

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 $70Z = 2^{3}\times3^{2}\times5^{3}\times7^{p+1}.$