√155²-134²/84

To simplify the expression \(\frac{\sqrt{155^2 - 134^2}}{84}\), we can first evaluate the term inside the square root.

Using the difference of squares formula, we have:

\[ a^2 - b^2 = (a - b)(a + b) \]

Here, let \(a = 155\) and \(b = 134\). Thus:

\[ 155^2 - 134^2 = (155 - 134)(155 + 134) \]

Calculating \(155 - 134\) and \(155 + 134\):

\[ 155 - 134 = 21 \] \[ 155 + 134 = 289 \]

Now substituting back into the expression:

\[ 155^2 - 134^2 = 21 \times 289 \]

Next, we need to calculate \(21 \times 289\):

\[ 21 \times 289 = 6069 \]

Now substituting back into the square root:

\[ \sqrt{155^2 - 134^2} = \sqrt{6069} \]

Next, we simplify the overall expression:

\[ \frac{\sqrt{6069}}{84} \]

To simplify \(\sqrt{6069}\), we can try to factor it. The prime factorization of \(6069\) is:

\[ 6069 = 3^2 \times 673 \]

This means:

\[ \sqrt{6069} = \sqrt{3^2 \times 673} = 3\sqrt{673} \]

Now substituting this back into our expression:

\[ \frac{3\sqrt{673}}{84} \]

Now we can simplify \(\frac{3}{84}\):

\[ \frac{3}{84} = \frac{1}{28} \]

Thus, the final expression is:

\[ \frac{\sqrt{673}}{28} \]

So, the final result is:

\[ \frac{\sqrt{673}}{28} \]