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

a.
`-( 4√(3) + 3)`

b.
`x^{(-5)/(12)} y^{(31)/(24)}`

c.
`(8√(x) + 8√(5))/(x - 5)`

d.
`45 + 12√(5y) + 4 y`

e.
`-((3)/(5x))^{(1)/(2)}`

Explanation:

a.

`√(12) - √(108) - \sqrt[3]{27}`

Expand each expression

`√(4*3) - √(36 * 3) - \sqrt[3]{3*3*3}`

Split the first two surds

`√(4)*√(3) - √(36) * √(3) - \sqrt[3]{3*3*3}`

`2*√(3) - 6 * √(3) - \sqrt[3]{3*3*3}`

Apply law of indices

`2*√(3) - 6 * √(3) - \sqrt[3]{3^3}`

Apply law of indices

`2*√(3) - 6 * √(3) - 3^{3*(1)/(3)}`

`2*√(3) - 6 * √(3) - 3^(1)`

`2*√(3) - 6 * √(3) - 3`

`2√(3) - 6√(3) - 3`

`- 4√(3) - 3`

Factorize

`-( 4√(3) + 3)`

The expression cannot be further simplified

b.

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

Expand the expression

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

Apply the following law of indices;

`(a^m)/(a^n) = a^(m-n)`

`x^{{(-3)/(4) - (-1)/(3)}} * y^{{(2)/(3) - (-5)/(8)}}}`

`x^{{(-3)/(4) + (1)/(3)}} * y^{{(2)/(3) + (5)/(8)}}}`

Add the exponents

`x^{(-9+4)/(12)} * y^{{(16+15)/(24)}}}`

`x^{(-5)/(12)} * y^{{(31)/(24)}}}`

`x^{(-5)/(12)} y^{(31)/(24)}`

The expression cannot be further simplified

c.

`(8)/(√(x) - √(5))`

Rationalize the denominator

`(8)/(√(x) - √(5)) * (√(x) + √(5))/(√(x) + √(5))`

`(8(√(x) + √(5)))/((√(x) - √(5))(√(x) + √(5)))`

Simplify the numerator

`(8√(x) + 8√(5))/((√(x) - √(5))(√(x) + √(5)))`

Simplify the denominator by difference of two squares

`(8√(x) + 8√(5))/(√(x)^2 - √(5)^2)`

`(8√(x) + 8√(5))/(x - 5)`

The expression cannot be further simplified

d.

`(3√(5) + 2√(y))^2`

Expand the expression

`(3√(5) + 2√(y))(3√(5) + 2√(y))`

Open the bracket

`3√(5) (3√(5) + 2√(y))+ 2√(y)(3√(5) + 2√(y))`

Open both brackets

`3√(5) *3√(5) + 3√(5)*2√(y)+ 2√(y)*3√(5) + 2√(y)*2√(y)`

`(3√(5) *3√(5)) + (3√(5)*2√(y))+ (2√(y)*3√(5)) + (2√(y)*2√(y))`

Multiply each expression in the bracket

`(3*3√(5*5)) + (3*2√(5*y))+ (2*3√(5*y)) + (2*2√(y*y))`

`(9√(25)) + (6√(5y))+ (6√(5y)) + (4√(y^2))`

Solve like terms

`(9√(25)) + (12√(5y)) + (4√(y^2))`

Take square root of 25 and y²

`(9 * 5) + (12√(5y)) + (4 * y)`

`(45) + (12√(5y)) + (4 y)`

Remove the brackets

`45 + 12√(5y) + 4 y`

The expression cannot be further simplified

e.

`-\sqrt{(3)/(5x)}`

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

`-((3)/(5x))^{(1)/(2)}`