Question

suppose n is an integer greater than 1. write an expression equivalent to \\(\sqrt{753^{2n}}\\) in simplest form.\\(\sqrt{753^{2n}} = \square\\)(type an exact answer, using radicals as needed.)

Explanation:

Step1: Recall the property of radicals

For any non - negative real number \(a\) and positive integer \(m\), \(\sqrt{a^{m}}=a^{\frac{m}{2}}\) when \(m\) is even (since we are dealing with square root here). Also, we know the exponent rule \((a^{b})^{c}=a^{bc}\).
We have the expression \(\sqrt{753^{2n}}\). Using the property of square roots \(\sqrt{x}=x^{\frac{1}{2}}\), we can rewrite the square root as an exponent: \((753^{2n})^{\frac{1}{2}}\).

Step2: Apply the exponent rule \((a^{m})^{n}=a^{mn}\)

When we have \((753^{2n})^{\frac{1}{2}}\), by the exponent rule \((a^{m})^{n}=a^{m\times n}\), here \(a = 753\), \(m=2n\) and \(n=\frac{1}{2}\). So we multiply the exponents: \(2n\times\frac{1}{2}=n\). So \((753^{2n})^{\frac{1}{2}}=753^{n}\).

Answer:

\(753^{n}\)