Answer:
a. [tex]-( 4\sqrt{3} + 3)[/tex]
b. [tex]x^{\frac{-5}{12}} y^{\frac{31}{24}}[/tex]
c. [tex]\frac{8\sqrt{x} + 8\sqrt{5}}{x - 5}[/tex]
d. [tex]45 + 12\sqrt{5y} + 4 y[/tex]
e. [tex]-(\frac{3}{5x})^{\frac{1}{2}}[/tex]
Step-by-step explanation:
a.
[tex]\sqrt{12} - \sqrt{108} - \sqrt[3]{27}[/tex]
Expand each expression
[tex]\sqrt{4*3} - \sqrt{36 * 3} - \sqrt[3]{3*3*3}[/tex]
Split the first two surds
[tex]\sqrt{4}*\sqrt{3} - \sqrt{36} * \sqrt{3} - \sqrt[3]{3*3*3}[/tex]
[tex]2*\sqrt{3} - 6 * \sqrt{3} - \sqrt[3]{3*3*3}[/tex]
Apply law of indices
[tex]2*\sqrt{3} - 6 * \sqrt{3} - \sqrt[3]{3^3}[/tex]
Apply law of indices
[tex]2*\sqrt{3} - 6 * \sqrt{3} - 3^{3*\frac{1}{3}}[/tex]
[tex]2*\sqrt{3} - 6 * \sqrt{3} - 3^{1}[/tex]
[tex]2*\sqrt{3} - 6 * \sqrt{3} - 3[/tex]
[tex]2\sqrt{3} - 6\sqrt{3} - 3[/tex]
[tex]- 4\sqrt{3} - 3[/tex]
Factorize
[tex]-( 4\sqrt{3} + 3)[/tex]
The expression cannot be further simplified
b.
[tex](\frac{x^{\frac{-3}{4}}y^{\frac{2}{3}}}{x^{\frac{-1}{3}}y^{\frac{-5}{8}}})[/tex]
Expand the expression
[tex](\frac{x^{\frac{-3}{4}} * y^{\frac{2}{3}}}{x^{\frac{-1}{3}} * y^{\frac{-5}{8}}})[/tex]
Apply the following law of indices;
[tex]\frac{a^m}{a^n} = a^{m-n}[/tex]
[tex]x^{{\frac{-3}{4} - \frac{-1}{3}}} * y^{{\frac{2}{3} - \frac{-5}{8}}}}[/tex]
[tex]x^{{\frac{-3}{4} + \frac{1}{3}}} * y^{{\frac{2}{3} + \frac{5}{8}}}}[/tex]
Add the exponents
[tex]x^{\frac{-9+4}{12}} * y^{{\frac{16+15}{24}}}}[/tex]
[tex]x^{\frac{-5}{12}} * y^{{\frac{31}{24}}}}[/tex]
[tex]x^{\frac{-5}{12}} y^{\frac{31}{24}}[/tex]
The expression cannot be further simplified
c.
[tex]\frac{8}{\sqrt{x} - \sqrt{5}}[/tex]
Rationalize the denominator
[tex]\frac{8}{\sqrt{x} - \sqrt{5}} * \frac{\sqrt{x} + \sqrt{5}}{\sqrt{x} + \sqrt{5}}[/tex]
[tex]\frac{8(\sqrt{x} + \sqrt{5})}{(\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5})}[/tex]
Simplify the numerator
[tex]\frac{8\sqrt{x} + 8\sqrt{5}}{(\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5})}[/tex]
Simplify the denominator by difference of two squares
[tex]\frac{8\sqrt{x} + 8\sqrt{5}}{\sqrt{x}^2 - \sqrt{5}^2}[/tex]
[tex]\frac{8\sqrt{x} + 8\sqrt{5}}{x - 5}[/tex]
The expression cannot be further simplified
d.
[tex](3\sqrt{5} + 2\sqrt{y})^2[/tex]
Expand the expression
[tex](3\sqrt{5} + 2\sqrt{y})(3\sqrt{5} + 2\sqrt{y})[/tex]
Open the bracket
[tex]3\sqrt{5} (3\sqrt{5} + 2\sqrt{y})+ 2\sqrt{y}(3\sqrt{5} + 2\sqrt{y})[/tex]
Open both brackets
[tex]3\sqrt{5} *3\sqrt{5} + 3\sqrt{5}*2\sqrt{y}+ 2\sqrt{y}*3\sqrt{5} + 2\sqrt{y}*2\sqrt{y}[/tex]
[tex](3\sqrt{5} *3\sqrt{5}) + (3\sqrt{5}*2\sqrt{y})+ (2\sqrt{y}*3\sqrt{5}) + (2\sqrt{y}*2\sqrt{y})[/tex]
Multiply each expression in the bracket
[tex](3*3\sqrt{5*5}) + (3*2\sqrt{5*y})+ (2*3\sqrt{5*y}) + (2*2\sqrt{y*y})[/tex]
[tex](9\sqrt{25}) + (6\sqrt{5y})+ (6\sqrt{5y}) + (4\sqrt{y^2})[/tex]
Solve like terms
[tex](9\sqrt{25}) + (12\sqrt{5y}) + (4\sqrt{y^2})[/tex]
Take square root of 25 and y²
[tex](9 * 5) + (12\sqrt{5y}) + (4 * y)[/tex]
[tex](45) + (12\sqrt{5y}) + (4 y)[/tex]
Remove the brackets
[tex]45 + 12\sqrt{5y} + 4 y[/tex]
The expression cannot be further simplified
e.
[tex]-\sqrt{\frac{3}{5x}}[/tex]
This expression can not be simplified; However, it can be rewritten, by applying law of indices as
[tex]-(\frac{3}{5x})^{\frac{1}{2}}[/tex]
