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ohaa [14]
3 years ago
9

Perform all indicated operations, and express each answer in simplest form with positive exponents. Assume that all variables re

present positive real numbers. Have to show all work.

Mathematics
1 answer:
Mademuasel [1]3 years ago
4 0

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)

<em>The expression cannot be further simplified</em>

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}}

<em>The expression cannot be further simplified</em>

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}

<em>The expression cannot be further simplified</em>

<em></em>

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

<em>The expression cannot be further simplified</em>

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}}

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