Question

Perform all indicated operations, and express each answer in simplest form with positive exponents. Assume that all variables represent positive real numbers. Have to show all work.

Answer 1

Answer:

a. $-( 4\sqrt{3} + 3)$

b.  $x^{\frac{-5}{12}} y^{\frac{31}{24}}$

c.  $\frac{8\sqrt{x} + 8\sqrt{5}}{x - 5}$

d. $45 + 12\sqrt{5y} + 4 y$

e. $-(\frac{3}{5x})^{\frac{1}{2}}$

Step-by-step explanation:

a.

$\sqrt{12} - \sqrt{108} - \sqrt[3]{27}$

Expand each expression

$\sqrt{4*3} - \sqrt{36 * 3} - \sqrt[3]{3*3*3}$

Split the first two surds

$\sqrt{4}*\sqrt{3} - \sqrt{36} * \sqrt{3} - \sqrt[3]{3*3*3}$

$2*\sqrt{3} - 6 * \sqrt{3} - \sqrt[3]{3*3*3}$

Apply law of indices

$2*\sqrt{3} - 6 * \sqrt{3} - \sqrt[3]{3^3}$

Apply law of indices

$2*\sqrt{3} - 6 * \sqrt{3} - 3^{3*\frac{1}{3}}$

$2*\sqrt{3} - 6 * \sqrt{3} - 3^{1}$

$2*\sqrt{3} - 6 * \sqrt{3} - 3$

$2\sqrt{3} - 6\sqrt{3} - 3$

$- 4\sqrt{3} - 3$

Factorize

$-( 4\sqrt{3} + 3)$

The expression cannot be further simplified

b.

$(\frac{x^{\frac{-3}{4}}y^{\frac{2}{3}}}{x^{\frac{-1}{3}}y^{\frac{-5}{8}}})$

Expand the expression

$(\frac{x^{\frac{-3}{4}} * y^{\frac{2}{3}}}{x^{\frac{-1}{3}} * y^{\frac{-5}{8}}})$

Apply the following law of indices;

$\frac{a^m}{a^n} = a^{m-n}$

$x^{{\frac{-3}{4} - \frac{-1}{3}}} * y^{{\frac{2}{3} - \frac{-5}{8}}}}$

$x^{{\frac{-3}{4} + \frac{1}{3}}} * y^{{\frac{2}{3} + \frac{5}{8}}}}$

Add the exponents

$x^{\frac{-9+4}{12}} * y^{{\frac{16+15}{24}}}}$

$x^{\frac{-5}{12}} * y^{{\frac{31}{24}}}}$

$x^{\frac{-5}{12}} y^{\frac{31}{24}}$

The expression cannot be further simplified

c.

$\frac{8}{\sqrt{x} - \sqrt{5}}$

Rationalize the denominator

$\frac{8}{\sqrt{x} - \sqrt{5}} * \frac{\sqrt{x} + \sqrt{5}}{\sqrt{x} + \sqrt{5}}$

$\frac{8(\sqrt{x} + \sqrt{5})}{(\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5})}$

Simplify the numerator

$\frac{8\sqrt{x} + 8\sqrt{5}}{(\sqrt{x} - \sqrt{5})(\sqrt{x} + \sqrt{5})}$

Simplify the denominator by difference of two squares

$\frac{8\sqrt{x} + 8\sqrt{5}}{\sqrt{x}^2 - \sqrt{5}^2}$

$\frac{8\sqrt{x} + 8\sqrt{5}}{x - 5}$

The expression cannot be further simplified

d.

$(3\sqrt{5} + 2\sqrt{y})^2$

Expand the expression

$(3\sqrt{5} + 2\sqrt{y})(3\sqrt{5} + 2\sqrt{y})$

Open the bracket

$3\sqrt{5} (3\sqrt{5} + 2\sqrt{y})+ 2\sqrt{y}(3\sqrt{5} + 2\sqrt{y})$

Open both brackets

$3\sqrt{5} *3\sqrt{5} + 3\sqrt{5}*2\sqrt{y}+ 2\sqrt{y}*3\sqrt{5} + 2\sqrt{y}*2\sqrt{y}$

$(3\sqrt{5} *3\sqrt{5}) + (3\sqrt{5}*2\sqrt{y})+ (2\sqrt{y}*3\sqrt{5}) + (2\sqrt{y}*2\sqrt{y})$

Multiply each expression in the bracket

$(3*3\sqrt{5*5}) + (3*2\sqrt{5*y})+ (2*3\sqrt{5*y}) + (2*2\sqrt{y*y})$

$(9\sqrt{25}) + (6\sqrt{5y})+ (6\sqrt{5y}) + (4\sqrt{y^2})$

Solve like terms

$(9\sqrt{25}) + (12\sqrt{5y}) + (4\sqrt{y^2})$

Take square root of 25 and y²

$(9 * 5) + (12\sqrt{5y}) + (4 * y)$

$(45) + (12\sqrt{5y}) + (4 y)$

Remove the brackets

$45 + 12\sqrt{5y} + 4 y$

The expression cannot be further simplified

e.

$-\sqrt{\frac{3}{5x}}$

This expression can not be simplified; However, it can be rewritten, by applying law of indices as

$-(\frac{3}{5x})^{\frac{1}{2}}$