Question

the square root function
quiz complete
1 2 3 4 5 6
what is the simplified form of \\(\sqrt{\frac{72x^{16}}{50x^{36}}}\\)? assume \\(x \
eq 0\\).
\\(\circ\\) \\(\frac{6}{5x^{10}}\\)
\\(\circ\\) \\(\frac{6}{5x^2}\\)
\\(\circ\\) \\(\frac{6}{5}x^{10}\\)
\\(\circ\\) \\(\frac{6}{5}x^2\\)
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Explanation:

Step1: Simplify the fraction inside the square root

First, simplify the fraction \(\frac{72x^{16}}{50x^{36}}\). We can simplify the coefficients and the variables separately. For the coefficients, \(\frac{72}{50}=\frac{36}{25}\) (dividing numerator and denominator by 2). For the variables, using the rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\), we have \(\frac{x^{16}}{x^{36}}=x^{16-36}=x^{-20}\). So now the expression inside the square root is \(\frac{36x^{-20}}{25}\).

Step2: Take the square root of the fraction

Now, take the square root of \(\sqrt{\frac{36x^{-20}}{25}}\). The square root of a fraction is the square root of the numerator divided by the square root of the denominator. So \(\sqrt{\frac{36x^{-20}}{25}}=\frac{\sqrt{36x^{-20}}}{\sqrt{25}}\). We know that \(\sqrt{36} = 6\) and \(\sqrt{25}=5\). For the variable part, \(\sqrt{x^{-20}}=x^{\frac{-20}{2}}=x^{-10}\) (using the rule \(\sqrt{a^m}=a^{\frac{m}{2}}\) for \(a\geq0\)). So now we have \(\frac{6x^{-10}}{5}\).

Step3: Rewrite the negative exponent

Recall that \(a^{-n}=\frac{1}{a^n}\), so \(x^{-10}=\frac{1}{x^{10}}\). Therefore, \(\frac{6x^{-10}}{5}=\frac{6}{5x^{10}}\).

Answer:

\(\frac{6}{5x^{10}}\) (which corresponds to the first option: \(\boldsymbol{\frac{6}{5x^{10}}}\))