Question 3

We will check if each number can be written as a simple fraction.

Step 1 — Classify $\sqrt{81}$

Let's look at the number $\sqrt{81}$. We know that $\mathbf{81}$ is a perfect square. It is the square of $\mathbf{9}$.

$\sqrt{81}$

$= \sqrt{9 \times 9}$

$= \sqrt{9^2}$

$\boxed{9}$

We can write $\mathbf{9}$ as $\frac{9}{1}$. This is a fraction of two integers. So, $\sqrt{81}$ is a rational number.

Step 2 — Classify $\sqrt{12}$

Let's look at the number $\sqrt{12}$. We can simplify the square root. We find the prime factors of $\mathbf{12}$.

$\sqrt{12}$

$= \sqrt{4 \times 3}$

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

$= 2\sqrt{3}$

The number $\sqrt{3}$ is not a whole number. It is a non-terminating and non-repeating decimal. So, $\sqrt{3}$ is an irrational number. When we multiply a rational number ($\mathbf{2}$) by an irrational number ($\sqrt{3}$), the result is irrational. So, $\sqrt{12}$ is an irrational number.

Step 3 — Classify $0.33333 \dots$

Let's look at the number $\mathbf{0.33333 \dots}$. This is a repeating decimal. Repeating decimals are always rational numbers. Let's convert it to a fraction. Let $x$ be the number.

$x = 0.33333 \dots \quad \text{(Equation 1)}$

Multiply by $\mathbf{10}$ to shift the decimal.

$10x = 3.33333 \dots \quad \text{(Equation 2)}$

Now, subtract Equation 1 from Equation 2.

$10x - x = 3.33333 \dots - 0.33333 \dots$

$9x = 3$

$x = \frac{3}{9}$

$\boxed{x = \frac{1}{3}}$

So, $\mathbf{0.33333 \dots}$ is a rational number.

Step 4 — Classify $0.123451234512345 \dots$

Let's look at the number $\mathbf{0.123451234512345 \dots}$. This is also a repeating decimal. The block of digits "$\mathbf{12345}$" repeats. Repeating decimals are always rational numbers. Let's convert it to a fraction. Let $x$ be the number.

$x = 0.1234512345 \dots \quad \text{(Equation 1)}$

The repeating block has $\mathbf{5}$ digits. Multiply by $\mathbf{10^5}$ (which is $\mathbf{100000}$).

$100000x = 12345.1234512345 \dots \quad \text{(Equation 2)}$

Now, subtract Equation 1 from Equation 2.

$100000x - x = 12345.12345 \dots - 0.12345 \dots$

$99999x = 12345$

$x = \frac{12345}{99999}$

We can simplify this fraction by dividing by $\mathbf{3}$.

$x = \frac{12345 \div 3}{99999 \div 3}$

$\boxed{x = \frac{4115}{33333}}$

So, $\mathbf{0.123451234512345 \dots}$ is a rational number.

Step 5 — Classify $1.01001000100001 \dots$

Let's look at the number $\mathbf{1.01001000100001 \dots}$. We see a pattern in the digits. There is one zero, then two zeros, then three zeros, and so on. The number of zeros between the ones keeps increasing. This means there is no fixed block of digits that repeats. The decimal expansion is non-terminating and non-repeating. Numbers with such decimal expansions are irrational. So, $\mathbf{1.01001000100001 \dots}$ is an irrational number.

Step 6 — Classify $23.560185612239874790120$

Let's look at the number $\mathbf{23.560185612239874790120}$. This decimal number ends after a certain number of digits. It is a terminating decimal. Terminating decimals are always rational numbers. We can write it as a fraction. Count the number of digits after the decimal point. There are $\mathbf{21}$ digits after the decimal point.

$23.560185612239874790120$

$\boxed{= \frac{23560185612239874790120}{1000000000000000000000}}$

So, $\mathbf{23.560185612239874790120}$ is a rational number.

Answer

(i) $\sqrt{81}$ is rational. Its fractional form is $\frac{9}{1}$. (ii) $\sqrt{12}$ is irrational. (iii) $0.33333 \dots$ is rational. Its fractional form is $\frac{1}{3}$. (iv) $0.123451234512345 \dots$ is rational. Its fractional form is $\frac{4115}{33333}$. (v) $1.01001000100001 \dots$ is irrational. (vi) $23.560185612239874790120$ is rational. Its fractional form is $\frac{23560185612239874790120}{1000000000000000000000}$.