Question

question
perform the operation and reduce the answer fully. make sure to express your answer as a simplified fraction.
\\(\frac{9}{\frac{7}{\frac{5}{2}}}\\)

Explanation:

The problem seems to be a fraction division or a complex fraction. Let's assume it's \(\frac{9}{\frac{7}{\frac{5}{2}}}\). First, we simplify the innermost fraction division. Dividing by a fraction is multiplying by its reciprocal. So \(\frac{7}{\frac{5}{2}} = 7\times\frac{2}{5}=\frac{14}{5}\). Then we have \(\frac{9}{\frac{14}{5}}\), which is \(9\times\frac{5}{14}=\frac{45}{14}\). But maybe the original problem is a vertical fraction stack, perhaps \(\frac{9}{7}\div\frac{5}{2}\)? Wait, the original input's formatting is unclear. Let's re - examine. If it's \(\frac{9}{7}\div\frac{5}{2}\), then dividing fractions: multiply by reciprocal. So \(\frac{9}{7}\times\frac{2}{5}=\frac{18}{35}\). But the original problem's display is a bit ambiguous. Wait, maybe it's a complex fraction like \(\frac{9}{\frac{7}{5/2}}\). Let's parse the original input again. The user wrote: \(\frac{9}{\frac{7}{\frac{5}{2}}}\). Let's compute from the bottom up. First, \(\frac{5}{2}\) is the bottom. Then \(\frac{7}{\frac{5}{2}}=7\times\frac{2}{5}=\frac{14}{5}\). Then \(\frac{9}{\frac{14}{5}} = 9\times\frac{5}{14}=\frac{45}{14}\). But maybe the problem is \(\frac{9}{7}\times\frac{5}{2}\)? No, the operation is not clear. Wait, the original problem says "Perform the operation and reduce the answer fully". Let's assume the operation is dividing \(\frac{9}{7}\) by \(\frac{5}{2}\) (maybe the vertical alignment was a fraction division). So:

Step 1: Recall the rule for dividing fractions

To divide two fractions \(\frac{a}{b}\div\frac{c}{d}\), we use the rule \(\frac{a}{b}\times\frac{d}{c}=\frac{a\times d}{b\times c}\), where \(b eq0\), \(c eq0\), \(d eq0\).

Step 2: Apply the rule

If we have \(\frac{9}{7}\div\frac{5}{2}\), then \(a = 9\), \(b = 7\), \(c = 5\), \(d = 2\). So we get \(\frac{9}{7}\times\frac{2}{5}=\frac{9\times2}{7\times5}\).

Step 3: Multiply the numerators and denominators

\(9\times2 = 18\) and \(7\times5=35\). So the result is \(\frac{18}{35}\). But wait, if the original problem was a complex fraction like \(\frac{9}{\frac{7}{5}\div2}\), let's check. \(\frac{7}{5}\div2=\frac{7}{5}\times\frac{1}{2}=\frac{7}{10}\), then \(\frac{9}{\frac{7}{10}}=9\times\frac{10}{7}=\frac{90}{7}\). But the original input's formatting is very unclear. Given the user's input: "9 over 7 over 5 over 2", which is a nested fraction: \(\frac{9}{\frac{7}{\frac{5}{2}}}\). Let's compute that:

First, \(\frac{5}{2}\) is the bottom. Then \(\frac{7}{\frac{5}{2}}=7\div\frac{5}{2}=7\times\frac{2}{5}=\frac{14}{5}\). Then \(\frac{9}{\frac{14}{5}}=9\div\frac{14}{5}=9\times\frac{5}{14}=\frac{45}{14}\). But maybe the problem was written as \(\frac{9}{7}\times\frac{5}{2}\)? No, the operation is "perform the operation", which is likely division. Given the ambiguity, but the most common nested fraction interpretation is \(\frac{9}{\frac{7}{\frac{5}{2}}}\). Let's re - do the calculation:

Step 1: Simplify the innermost fraction division

\(\frac{5}{2}\) is the denominator of the middle fraction. So \(\frac{7}{\frac{5}{2}}=7\times\frac{2}{5}=\frac{14}{5}\) (because dividing by a fraction is multiplying by its reciprocal).

Step 2: Simplify the outer fraction division

Now we have \(\frac{9}{\frac{14}{5}}\), which is \(9\div\frac{14}{5}=9\times\frac{5}{14}=\frac{45}{14}\) (again, dividing by a fraction is multiplying by its reciprocal).

But maybe the problem was a simple fraction division of \(\frac{9}{7}\) and \(\frac{5}{2}\), i.e., \(\frac{9}{7}\div\frac{5}{2}\). Let's calculate that:

Step 1: Use the fraction division rule

\(\frac{9}{7}\div\frac{5}{2}=\frac{9}{7}\times\frac{2}{5}\) (since dividing by \(\frac{5}{2}\) is multiplying by \(\frac{2}{5}\)).

Step 2: Multiply the numerators and denominators

\(9\times2 = 18\) and \(7\times5 = 35\), so \(\frac{18}{35}\).

Given that the user's input formatting is not clear, but if we assume the operation is \(\frac{9}{7}\div\frac{5}{2}\) (maybe a typo in the nested fraction), the result is \(\frac{18}{35}\). If it's the nested fraction \(\frac{9}{\frac{7}{\frac{5}{2}}}\), the result is \(\frac{45}{14}\). Since the original input's formatting is "9 7 5 2" in a vertical fraction stack, the most likely interpretation is a nested fraction \(\frac{9}{\frac{7}{\frac{5}{2}}}\). Let's confirm the order of operations for nested fractions: we start from the bottom. So \(\frac{5}{2}\) is the bottom, then \(\frac{7}{\frac{5}{2}}\), then \(\frac{9}{\frac{7}{\frac{5}{2}}}\).

Calculating \(\frac{7}{\frac{5}{2}}\):
Dividing by a fraction is multiplying by its reciprocal, so \(\frac{7}{\frac{5}{2}}=7\times\frac{2}{5}=\frac{14}{5}\)

Then calculating \(\frac{9}{\frac{14}{5}}\):
\(\frac{9}{\frac{14}{5}}=9\times\frac{5}{14}=\frac{45}{14}\)

Answer:

\(\frac{45}{14}\) (If the operation is the nested fraction division) or \(\frac{18}{35}\) (if it's \(\frac{9}{7}\div\frac{5}{2}\)). Given the original input's formatting as a vertical stack of 9, 7, 5, 2 (nested fraction), the answer is \(\frac{45}{14}\)