Questions: Simplify: 8 √128

Solution Steps

To simplify \(8 \sqrt{128}\), we first simplify the square root of 128. We can do this by finding the prime factorization of 128 and then simplifying the square root by taking out pairs of prime factors. After simplifying the square root, we multiply the result by 8.

We start with the expression \(8 \sqrt{128}\). To simplify \(\sqrt{128}\), we find its prime factorization: \[ 128 = 2^7 \] Thus, we can express the square root as: \[ \sqrt{128} = \sqrt{2^7} = 2^{7/2} = 2^3 \cdot \sqrt{2} = 8 \sqrt{2} \]

Now, we substitute back into the original expression: \[ 8 \sqrt{128} = 8 \cdot (8 \sqrt{2}) = 64 \sqrt{2} \]

To find the numerical value of \(64 \sqrt{2}\), we use the approximate value of \(\sqrt{2} \approx 1.4142\): \[ 64 \sqrt{2} \approx 64 \cdot 1.4142 \approx 90.5097 \]

Final Answer