Express the following as a rational number.
`((21)/(89))^2`

Answer

To express \(\left(\frac{21}{89}\right)^2\) as a rational number, we can follow these steps:

### Step 1: Write the expression in terms of squares
We start with the expression:
\[ \left(\frac{21}{89}\right)^2 = \frac{21^2}{89^2} \]

### Step 2: Calculate \(21^2\)
Next, we calculate \(21^2\):
\[ 21^2 = 21 \times 21 = 441 \]

### Step 3: Calculate \(89^2\)
Now, we calculate \(89^2\):
\[ 89^2 = 89 \times 89 = 7921 \]

### Step 4: Write the final expression
Now that we have both squares, we can write the expression as:
\[ \frac{21^2}{89^2} = \frac{441}{7921} \]

### Step 5: Verify the rational number form
We check if \(\frac{441}{7921}\) is in the form \(\frac{P}{Q}\) where \(Q \neq 0\). Here, \(P = 441\) and \(Q = 7921\), and since \(7921 \neq 0\), it confirms that it is a rational number.

### Final Answer
Thus, the expression \(\left(\frac{21}{89}\right)^2\) as a rational number is:
\[ \frac{441}{7921} \]

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