Let's analyze each expression to determine if it is rational or irrational.
### Expression 1: [tex]\( 7.5 \overline{1} \cdot (-4) \)[/tex]
This expression involves the multiplication of a repeating decimal [tex]\( 7.5 \overline{1} \)[/tex] with [tex]\( -4 \)[/tex].
Since [tex]\( 7.5 \overline{1} \)[/tex] is a repeating decimal, it is a rational number, because all repeating decimals are rational numbers. Multiplying a rational number by an integer (which is also rational) results in another rational number. Thus, we know the product [tex]\( 7.5 \overline{1} \cdot (-4) \)[/tex] is rational.
### Expression 2: [tex]\( \sqrt{16} + \frac{3}{4} \)[/tex]
First, compute [tex]\( \sqrt{16} \)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
Next, add [tex]\( \frac{3}{4} \)[/tex] to [tex]\( 4 \)[/tex]:
[tex]\[ 4 + \frac{3}{4} = 4 + 0.75 = 4.75 \][/tex]
Since [tex]\( 4.75 \)[/tex] is a terminating decimal, it is a rational number.
### Expression 3: [tex]\( \sqrt{3} + 8.486 \)[/tex]
First, [tex]\( \sqrt{3} \)[/tex] is an irrational number because it cannot be expressed as a fraction of two integers.
Adding an irrational number ([tex]\( \sqrt{3} \)[/tex]) to any number (in this case, the terminating decimal [tex]\( 8.486 \)[/tex]) results in an irrational number. Thus, [tex]\( \sqrt{3} + 8.486 \)[/tex] is irrational.
### Expression 4: [tex]\( 8 \frac{2}{3} \times 17.75 \)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 8 \frac{2}{3} = 8 + \frac{2}{3} = \frac{24}{3} + \frac{2}{3} = \frac{26}{3} \][/tex]
Now, perform the multiplication:
[tex]\[ \frac{26}{3} \times 17.75 \][/tex]
Since 17.75 is a terminating decimal, it can be expressed as a fraction, [tex]\( \frac{1775}{100} \)[/tex].
Thus:
[tex]\[ \frac{26}{3} \times \frac{1775}{100} \][/tex]
Performing the multiplications and simplifying will result in a rational number, as the product of two rational numbers is also rational.
### Conclusion
From the analysis, [tex]\( \sqrt{3} + 8.486 \)[/tex] is the irrational expression among the given ones.
