Write $\sqrt{2 \frac{2}{9}}$ in the form $\frac{p}{q} \sqrt{r}$ where $p$ and $q$ and $r$ are integers.
Answer (2)
Convert the mixed number to an improper fraction: 2 9 2 = 9 20 .
Separate the square root: 9 20 = 9 20 .
Simplify the numerator and denominator: 20 = 2 5 and 9 = 3 .
Write the result in the required form: 3 2 5 . The final answer is 3 2 5 .
Explanation
Understanding the Problem We are given the expression 2 9 2 and we want to write it in the form q p r , where p , q , and r are integers.
Converting to an Improper Fraction First, we need to convert the mixed number 2 9 2 to an improper fraction. To do this, we multiply the whole number part (2) by the denominator (9) and add the numerator (2). This gives us 2 ⋅ 9 + 2 = 18 + 2 = 20 . So, the improper fraction is 9 20 .
Substituting into the Square Root Now we substitute this improper fraction into the square root: 2 9 2 = 9 20 .
Separating the Square Root Next, we simplify the square root by separating the numerator and denominator: 9 20 = 9 20 .
Simplifying Numerator and Denominator We simplify the numerator and denominator separately. For the numerator, we have 20 = 4 ⋅ 5 = 4 ⋅ 5 = 2 5 . For the denominator, we have 9 = 3 .
Substituting Back Now we substitute these simplified expressions back into the fraction: 9 20 = 3 2 5 .
Final Form Finally, we write the result in the form q p r : 3 2 5 = 3 2 5 . Here, p = 2 , q = 3 , and r = 5 , which are all integers.
Conclusion Thus, the expression 2 9 2 in the form q p r is 3 2 5 .
Examples
Imagine you're calculating the length of the diagonal of a rectangle. If the area of the rectangle is expressed as a mixed number under a square root, converting it to the form q p r simplifies the calculation. This skill is also useful in physics when dealing with velocities or accelerations that involve square roots of fractions. Simplifying these expressions makes it easier to understand and apply the formulas correctly.
The expression 2 9 2 can be rewritten in the form q p r as 3 2 5 . This is achieved by converting the mixed number to an improper fraction, simplifying the square root, and separating the terms. The final result shows p = 2 , q = 3 , and r = 5 .
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