How do I do this Surd question?

Question

Write:

√(18k) x (√(8k - 2k) / (√54 - √24)

In its simplest form. Thanks

Answer 1

Given that,

                  √18k  X (√(8k - 2k) / (√54 - √24))

            => √(9 X 2)k  X (√6k / (√(9 X 6) - √(4 X 6))  [split 18 = 9X2,  54 = 9X6, 24 = 4X6]

            =>  3√2k  X  (√6k / (3√6 - 2√6))                    [since √9 = 3, √4 = 2]

            =>  3√2k  X  (√6k / √6)

            => 3√2k  X  √k

            => 3k√2.

This is the simplest form.

Answer 2

The surd expression is [ √(18k) · √(8k - 2k ) ] / [√(54) - √(24) ].

= [ √(18k) · √(6k ) ] / [√(54) - √(24) ]

Apply product property of radical :

1. If n is even and a and b are both nonnegative, then and

2. If n is odd, then .

= [ √(18k · 6k ) ] / [√(54) - √(24) ]

Factor into squares where possible.

= [ √(6² · k² · 3 ) ] / [√(3² · 6) - √(2² · 6) ]

= [ √6² · √k² · √3 ] / [ √3² · √6 - √2² · √6 ]

= [ 6 · k · √3 ] / [ 3 · √6 - 2 · √6 ]

= [ 6k√3 ] / [ (3 - 2) · √6 ]

= [ 6k√3 ] / [ √6 ]

= √6 · k · √3

= k√(18)

= k[ √3² · √2 ]

= 3k√2.