Some days ago, i've seen on reddit a post about the text of an admission exam to the MIT from 1869-1870.

I wanted to brush up on my algebra skills for a while, and I took this opportunity to try solving it.

Which exercises i solved

1. If \(e = 8\), find the numerical value of the following expression: \(e - \left\lbrace \sqrt{(e + 1)} + 2 \right\rbrace + \left(e - \sqrt[3]{e}\right) \sqrt{(e - 4)}. \)

\[ = 8 - \left\lbrace \sqrt{(8 + 1)} + 2 \right\rbrace + \left(8 - \sqrt[3]{8}\right) \sqrt{(8 - 4)} \]

\[ = 8 - \left\lbrace \sqrt{9} + 2 \right\rbrace + (8 - 2) \sqrt{4}\]

\[ = 8 - (3 + 2) + 6(2)\]

\[ = 8 - 5 + 12\]

\[ = 15\]

\[ \]

2. Simplify the following expression by removing the brackets and collecting like terms: \( 3a - \left[ b + \left( 2a - b \right) - \left( a - b \right) \right]. \)

\[ = 3a-\left\lbrack b+2a-b-a+b\right\rbrack\]

\[ = 3a-b-2a+b+a-b\]

\[ =2a-b\]

\[ \]

5. Simplify \(\left\lbrace\frac{a+b}{a-b}+\frac{a-b}{a+b}\right\rbrace\div\left\lbrace\frac{a+b}{a-b}-\frac{a-b}{a+b}\right\rbrace\).

\[ =\frac{\left(a+b)(a+b\right)+(a-b)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}\div\frac{\left(a+b)(a+b\right)-(a-b)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}\]

\[ =\frac{\left(a+b)(a+b\right)+(a-b)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}\frac{\left(a+b\right)\left(a-b\right)}{\left(a+b)(a+b\right)-(a-b)\left(a-b\right)}\]

\[ =\frac{\left(a+b)(a+b\right)+(a-b)\left(a-b\right)}{1}\frac{1}{\left(a+b)(a+b\right)-(a-b)\left(a-b\right)}\]

\[ =\frac{\left(a+b)(a+b\right)+(a-b)\left(a-b\right)}{\left(a+b)(a+b\right)-(a-b)\left(a-b\right)}\]

\[ =\frac{a^{2}+b^{2}+2ab+a^{2}+b^{2}-2ab}{a^{2}+b^{2}+2ab-\left(a^{2}+b^{2}-2ab\right)}\]

\[ =\frac{a^{2}+b^{2}+2ab+a^{2}+b^{2}-2ab}{a^{2}+b^{2}+2ab-a^{2}-b^{2}+2ab}\]

\[ =\frac{2a^{2}+2b^{2}}{4ab}\]

\[ =\frac{a^{2}+b^{2}}{2ab}\]

\[ \]

6. Solve \(\frac{3x-4}{2}-\frac{6x-5}{8}=\frac{3x-1}{16}\).

\[ \frac{8\left(3x-4\right)-2\left(6x-5\right)-3x+1}{16}=0\]

\[ \frac{24x-32-12x+10-3x+1}{16}=0\]

\[ \frac{12x-21}{16}=0\]

Multiplying both sides by 16 yields:

\[ 12x-21=0\]

\[ 12x=21\]

\[ x=\frac{21}{12}\]

Simplifying:

\[ x=\frac{7}{4}\]

\[ \]

7. Solve \(7x-5y=24, \qquad 4x-3y=11\).

a)

\[ 7x=5y+24 \qquad x=\frac{5}{7}y+\frac{24}{7}\]

\[ 5y=7x-24 \qquad y=\frac{7}{5}x-\frac{24}{5}\]

b)

\[ 4x=3y+11 \qquad x=\frac{3}{4}y+\frac{11}{4}\]

\[ 3y=4x-11 \qquad x=\frac{4}{3}x-\frac{11}{3}\]

\[ \]

Which exercise i couldn't solve

3. Multiply \(3{a}^2+ab-b^2\) by \(a^2-2ab-3b^2\), and divide the result by \(a+b\).

4. Reduce the following expression to its lowest terms: \(\frac{x^{6}+a^{2}x^{3}y}{x^{6}-a^{4}y^{2}}\).

Conclusion

These mathemathical expressions are written in lateX, interpreted and visualized by plugin MathJax. You have to learn the sintax, otherwise there is a graphical editorto convert expressions, written or imported from an image, to lateX language. I'm really happy for the accomplished result even if i solved only 5 assignments of 7. Last time i've studied Maths was 10 years ago. To verify the solutions i advice you to use wolframalpha.com. I invite willing readers to try to solve the unsolved exercises. I'd love to read step by step solutions at the email you can find in here.