Understanding Rational and Irrational Numbers

To determine which of the given numbers is irrational, we first need to understand the difference between rational and irrational numbers.

• Rational Numbers: These can be expressed in the form of a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not zero. Their decimal expansions are either terminating (like 0.5) or repeating (like 0.333...).
• Irrational Numbers: These cannot be expressed in the form of a fraction $\frac{p}{q}$. Their decimal expansions are non-terminating and non-repeating (like $\sqrt{2} \approx 1.41421356...$ or $\pi \approx 3.14159265...$).

Let's examine each option to see if it is a rational or an irrational number.

Analyzing Option 1: 3.3333 .....

The number $3.3333 .....$ is a decimal that repeats the digit '3' infinitely. A repeating decimal is always a rational number.

We can express it as a fraction:

Let $x = 3.3333 .....$

Multiply by 10: $10x = 33.3333 .....$

Subtract the first equation from the second:

$$10x - x = 33.3333... - 3.3333...$$ $$9x = 30$$ $$x = \frac{30}{9} = \frac{10}{3}$$

Since $3.3333 .....$ can be written as the fraction $\frac{10}{3}$, it is a rational number.

Analyzing Option 2: 5.63636363 .....

The number $5.63636363 .....$ is a decimal that repeats the block '63' infinitely. This is also a repeating decimal, and therefore, a rational number.

We can express it as a fraction:

Let $x = 5.63636363 .....$

Multiply by 100 (since two digits repeat): $100x = 563.636363 .....$

Subtract the first equation from the second:

$$100x - x = 563.6363... - 5.6363...$$ $$99x = 558$$ $$x = \frac{558}{99}$$

Since $5.63636363 .....$ can be written as the fraction $\frac{558}{99}$, it is a rational number.

Analyzing Option 3: (81) ^½ × (18) ^-½

Let's simplify this expression step by step using the properties of exponents and radicals.

• $(81)^{1/2}$ is the square root of 81. $\sqrt{81} = 9$.
• $(18)^{-1/2}$ means $\frac{1}{(18)^{1/2}}$, which is $\frac{1}{\sqrt{18}}$.

So the expression becomes:

$$9 \times \frac{1}{\sqrt{18}} = \frac{9}{\sqrt{18}}$$

We can simplify $\sqrt{18}$:

$$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$$

Substitute this back into the expression:

$$\frac{9}{3\sqrt{2}} = \frac{3}{\sqrt{2}}$$

To remove the radical from the denominator (rationalize), multiply by $\frac{\sqrt{2}}{\sqrt{2}}$:

$$\frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}$$

The number is $\frac{3\sqrt{2}}{2}$. Since $\sqrt{2}$ is an irrational number, and multiplying or dividing a non-zero rational number (like $\frac{3}{2}$) by an irrational number results in an irrational number, $\frac{3\sqrt{2}}{2}$ is an irrational number.

Analyzing Option 4: (256) ^¼ × (16) ^-¼

Let's simplify this expression.

• $(256)^{1/4}$ is the fourth root of 256. We know $4^4 = 4 \times 4 \times 4 \times 4 = 16 \times 16 = 256$. So, $\sqrt[4]{256} = 4$.
• $(16)^{-1/4}$ means $\frac{1}{(16)^{1/4}}$, which is $\frac{1}{\sqrt[4]{16}}$. We know $2^4 = 2 \times 2 \times 2 \times 2 = 16$. So, $\sqrt[4]{16} = 2$.

So the expression becomes:

$$4 \times \frac{1}{2} = \frac{4}{2} = 2$$

The number is 2. Since 2 can be written as $\frac{2}{1}$, it is a rational number.

Conclusion

Based on the analysis of each option:

• Option 1: $3.3333 .....$ is Rational.
• Option 2: $5.63636363 .....$ is Rational.
• Option 3: $(81)^{1/2} \times (18)^{-1/2} = \frac{3\sqrt{2}}{2}$ is Irrational.
• Option 4: $(256)^{1/4} \times (16)^{-1/4} = 2$ is Rational.

Therefore, the number that is irrational is the one presented in Option 3.