To determine which expression represents an irrational number, let's evaluate each option step-by-step:
Option A: [tex]\( \sqrt{169} \)[/tex]
Calculate [tex]\( \sqrt{169} \)[/tex]:
[tex]\[ \sqrt{169} = 13 \][/tex]
Since 13 is a whole number, it is a rational number.
Option B: [tex]\( \sqrt{400-100} \)[/tex]
First, compute the expression inside the square root:
[tex]\[ 400 - 100 = 300 \][/tex]
Now, take the square root of 300:
[tex]\[ \sqrt{300} \][/tex]
The number 300 is not a perfect square, so [tex]\( \sqrt{300} \)[/tex] cannot be expressed as a ratio of two integers. Therefore, it is an irrational number.
Option C: [tex]\( \sqrt{\frac{225}{144}} \)[/tex]
Simplify [tex]\( \frac{225}{144} \)[/tex]:
[tex]\[ \sqrt{\frac{225}{144}} = \frac{\sqrt{225}}{\sqrt{144}} = \frac{15}{12} = \frac{5}{4} \][/tex]
Since [tex]\( \frac{5}{4} \)[/tex] is a ratio of two integers, it is a rational number.
Option D: [tex]\( \sqrt{67-31} \)[/tex]
First, compute the expression inside the square root:
[tex]\[ 67 - 31 = 36 \][/tex]
Now, take the square root of 36:
[tex]\[ \sqrt{36} = 6 \][/tex]
Since 6 is a whole number, it is a rational number.
Option E: [tex]\( \sqrt{3} \cdot \sqrt{27} \)[/tex]
Simplify the product of the square roots:
[tex]\[ \sqrt{3} \cdot \sqrt{27} = \sqrt{3 \cdot 27} = \sqrt{81} = 9 \][/tex]
Since 9 is a whole number, it is a rational number.
Among the given options, the only expression that represents an irrational number is:
[tex]\[ B. \sqrt{400-100} = \sqrt{300} \approx 17.320508075688775 \][/tex]
Thus, the correct option is [tex]\( B \)[/tex].
