B3. For $x>0$ simplify the expression $(\sqrt{7}+2) \cdot \sqrt{11 \sqrt{x}-4 \sqrt{7 x}}+\sqrt[3]{27 x^{-3}+27 x^{-2}+9 x^{-1}+1}-\frac{3 x}{\sqrt[4]{x^{3}}}$.

## 22nd Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools National Competition, April 23, 2022

## Problems for 3rd Year

Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Enter the answers for section A in the left table, leave the right table blank.
![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-10.jpg?height=168&width=804&top_left_y=590&top_left_x=638)
B3. First, we write $(\sqrt{7}+2)$ under the root as $(\sqrt{7}+2)=\sqrt{(\sqrt{7}+2)^{2}}=\sqrt{11+4 \sqrt{7}}$, then we write $\sqrt{11 \sqrt{x}-4 \sqrt{7 x}}$ as $\sqrt{(11-4 \sqrt{7}) \cdot \sqrt{x}}$ and calculate $(\sqrt{7}+2) \cdot \sqrt{11 \sqrt{x}-4 \sqrt{7 x}}=$ $\sqrt{(11+4 \sqrt{7}) \cdot(11-4 \sqrt{7}) \cdot \sqrt{x}}=3 \cdot \sqrt[4]{x}$. Next, we calculate the value of the expression $\sqrt[3]{27 x^{-3}+27 x^{-2}+9 x^{-1}+1}=$ $\sqrt[3]{\frac{27+27 x+9 x^{2}+x^{3}}{x^{3}}}=\sqrt[3]{\frac{(x+3)^{3}}{x^{3}}}=\frac{x+3}{x}$. Then we rationalize the fraction $\frac{3 x}{\sqrt[4]{x^{3}}}$ and get $\frac{3 x}{\sqrt[4]{x^{3}}}=3 \cdot \sqrt[4]{x}$. When we add all the terms, we get the value of the desired expression, which is equal to $\frac{x+3}{x}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-21.jpg?height=80&width=1756&top_left_y=928&top_left_x=161)

Multiplication under the root ........................................................................................

Calculation of the expression $(\sqrt{7}+2) \cdot \sqrt{11 \sqrt{x}-4 \sqrt{7 x}}=3 \cdot \sqrt[4]{x} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .1$ point

Considering that $\sqrt[3]{27 x^{-3}+27 x^{-2}+9 x^{-1}+1}=\sqrt[3]{\frac{27+27 x+9 x^{2}+x^{3}}{x^{3}}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots .1$ point Writing the expression $27+27 x+9 x^{2}+x^{3}$ as a cube of a sum $(x+3)^{3} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point Calculation of the cube root $\sqrt[3]{27 x^{-3}+27 x^{-2}+9 x^{-1}+1}=\sqrt[3]{\frac{(x+3)^{3}}{x^{3}}}=\frac{x+3}{x} \ldots \ldots \ldots \ldots \ldots .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-21.jpg?height=72&width=1753&top_left_y=1329&top_left_x=163)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-21.jpg?height=72&width=1751&top_left_y=1386&top_left_x=164)

## 22nd Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 23, 2022  Solutions to Problems for 3rd Year

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-22.jpg?height=137&width=234&top_left_y=477&top_left_x=928)