Question

write a simplified expression that is equivalent to ((5x^{3})^{2}(9x^{8})^{1/2})

Explanation:

Step1: Apply power of a product rule

For \((5x^{3})^{2}\), using \((ab)^n = a^n b^n\), we get \(5^{2}(x^{3})^{2}\). For \((9x^{8})^{1/2}\), we get \(9^{1/2}(x^{8})^{1/2}\).
So the expression becomes \(5^{2}(x^{3})^{2}\times9^{1/2}(x^{8})^{1/2}\).

Step2: Simplify each power

Calculate \(5^{2}=25\), \((x^{3})^{2}=x^{3\times2}=x^{6}\), \(9^{1/2}=\sqrt{9} = 3\), \((x^{8})^{1/2}=x^{8\times\frac{1}{2}}=x^{4}\).
Now the expression is \(25\times x^{6}\times3\times x^{4}\).

Step3: Multiply the coefficients and combine like bases

Multiply the coefficients \(25\times3 = 75\). For the \(x\) terms, use \(a^m\times a^n=a^{m + n}\), so \(x^{6}\times x^{4}=x^{6 + 4}=x^{10}\).
Putting it together, we get \(75x^{10}\).

Answer:

\(75x^{10}\)