The number \( \left(4^{10}-4^{8}\right)\left(5^{10}-5^{8}\right)\left(9^{10}-9^{8}\right) \) is equal to \( 2^{x} \times 3^{y} \times 5^{z} \). Determine the value of \( x+y+z \)

To solve the expression \( \left(4^{10}-4^{8}\right)\left(5^{10}-5^{8}\right)\left(9^{10}-9^{8}\right) \), we will simplify each term separately. 1. **Simplifying \( 4^{10} - 4^{8} \)**: \[ 4^{10} - 4^{8} = 4^{8}(4^{2} - 1) = 4^{8}(16 - 1) = 4^{8} \cdot 15 \] Next, we express \( 4^{8} \) in terms of its prime factors: \[ 4^{8} = (2^2)^{8} = 2^{16} \] Thus, \[ 4^{10} - 4^{8} = 2^{16} \cdot 15 \] 2. **Simplifying \( 5^{10} - 5^{8} \)**: \[ 5^{10} - 5^{8} = 5^{8}(5^{2} - 1) = 5^{8}(25 - 1) = 5^{8} \cdot 24 \] We can factor \( 24 \) as: \[ 24 = 2^3 \cdot 3 \] Therefore, \[ 5^{10} - 5^{8} = 5^{8} \cdot (2^3 \cdot 3) = 5^{8} \cdot 2^3 \cdot 3 \] 3. **Simplifying \( 9^{10} - 9^{8} \)**: \[ 9^{10} - 9^{8} = 9^{8}(9^{2} - 1) = 9^{8}(81 - 1) = 9^{8} \cdot 80 \] We can express \( 9^{8} \) in terms of its prime factors: \[ 9^{8} = (3^2)^{8} = 3^{16} \] And \( 80 \) can be factored as: \[ 80 = 2^4 \cdot 5 \] Thus, \[ 9^{10} - 9^{8} = 3^{16} \cdot (2^4 \cdot 5) = 3^{16} \cdot 2^4 \cdot 5 \] Now we combine all the simplified terms: \[ (4^{10} - 4^{8})(5^{10} - 5^{8})(9^{10} - 9^{8}) = (2^{16} \cdot 15)(5^{8} \cdot 2^3 \cdot 3)(3^{16} \cdot 2^4 \cdot 5) \] Next, we will express \( 15 \) in terms of its prime factors: \[ 15 = 3 \cdot 5 \] Thus, \[ 4^{10} - 4^{8} = 2^{16} \cdot (3 \cdot 5) = 2^{16} \cdot 3^1 \cdot 5^1 \] Now we can write the entire product: \[ = (2^{16} \cdot 3^1 \cdot 5^1)(5^{8} \cdot 2^3 \cdot 3)(3^{16} \cdot 2^4 \cdot 5) \] Combining the powers of each prime factor: - For \( 2 \): \[ 2^{16} \cdot 2^3 \cdot 2^4 = 2^{16 + 3 + 4} = 2^{23} \] - For \( 3 \): \[ 3^1 \cdot 3^1 \cdot 3^{16} = 3^{1 + 1 + 16} = 3^{18} \] - For \( 5 \): \[ 5^1 \cdot 5^8 \cdot 5^1 = 5^{1 + 8 + 1} = 5^{10} \] Thus, we have: \[ (4^{10} -