B4. Simplify the expression $\frac{1}{a}\left(1+\frac{1}{a^{2}}\right)^{-\frac{1}{2}} \cdot\left(1+a^{2}\right)^{\frac{1}{2}}+6.25^{-\frac{1}{2}} \cdot 0.008^{-\frac{2}{3}}$ without using a calculator.

(7 points)

Space for solving tasks of set $B$.

## dMFA

12th competition in mathematics knowledge for students of secondary technical and vocational schools

Regional competition, March 28, 2012

## PROBLEMS FOR THIRD YEAR STUDENTS

Time for solving: 90 minutes. In set A, we will evaluate a correct answer with two points, while we will deduct half a point for an incorrect answer. Write the answers of set A in the left table.
![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-06.jpg?height=168&width=1194&top_left_y=660&top_left_x=446)
B4. The expression in the first parentheses $1+\frac{1}{a^{2}}$ is simplified to $\frac{a^{2}+1}{a^{2}}$. Considering the definition of a power with a rational exponent, we simplify the expression $\frac{1}{a}\left(\frac{a^{2}+1}{a^{2}}\right)^{-\frac{1}{2}}=\frac{a}{a \sqrt{a^{2}+1}}$. We simplify both parentheses to $\frac{a}{a \sqrt{a^{2}+1}} \cdot \sqrt{1+a^{2}}=1$. We calculate $6.25^{-\frac{1}{2}}=\frac{2}{5}$ and $0.008^{-\frac{2}{3}}=25$. We calculate $\frac{2}{5} \cdot 25=10$. We add and get the result 11.

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-14.jpg?height=74&width=1685&top_left_y=1679&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-14.jpg?height=69&width=1687&top_left_y=1736&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-14.jpg?height=55&width=1685&top_left_y=1800&top_left_x=228)
Simplified expression $\frac{1}{a}\left(1+\frac{1}{a^{2}}\right)^{-\frac{1}{2}}=\frac{a}{a \sqrt{a^{2}+1}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point Simplification of the first term $\frac{a}{a \sqrt{a^{2}+1}} \cdot \sqrt{1+a^{2}}=1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-14.jpg?height=68&width=1682&top_left_y=1982&top_left_x=230)
Result 11 ..........................................................................................................

## Third Year

## Group A

| A1 | A2 | A3 | A4 | A5 | A6 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| A | A | E | B | E | E |