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}
]
