a. √3(√6 +3√8-6√7)
= √3√6 + 3√3√8 - 6√3√7
= √(3*6) + 3√(3*8) - 6√(3*7)
= √18 + 3√24 - 6√21
= √2 * √9 + 3√2 * √6 - 2√3 * √7
= 3√2 + 3√2√6 - 2√3√7
= 3√2 + 3√12 - 2√21
= 3√2 + 3√(2*2*3) - 2√21
= 3√2 + 6√3 - 2√21
b. (2 + √5)(2-√5)
= 2*2 + 2*(-√5) + 2√5 + (√5)(-√5)
= 4 - 2√5 + 2√5 - 5
= -1
c. 2³√3 x ³√72 - ⁴√125 x ⁴√5
= (2³√3 x ³√(2*2*2*3) - ⁴√(5*5*5) x ⁴√5
= 2³√3 x ³√(2³√3) - ⁴√(5⁴√5)
= 2³√3 x 2√3 - 5√5
= 2^(3+1)√(3*3) - 5√5
= 2^4 * 3 - 5√5
= 16√3 - 5√5
d. (4√27 + 3√2)(4√3 - √2)
= (4√(3*3*3) + 3√2)(4√3 - √2)
= 4√3 * 3 + 3√2 * 4√3 - 4√3 * √2 - 3√2 * √2
= 12√3 + 12√(2*3) - 4√(3*2) - 3√(2*2)
= 12√3 + 12√6 - 4√6 - 3√4
= 12√3 + 12√6 - 4√6 - 3 * 2
= 12√3 + 12√6 - 4√6 - 6
= 12√3 + 8√6 - 6
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a. √3(√6 +3√8-6√7)
= √3√6 + 3√3√8 - 6√3√7
= √(3*6) + 3√(3*8) - 6√(3*7)
= √18 + 3√24 - 6√21
= √2 * √9 + 3√2 * √6 - 2√3 * √7
= 3√2 + 3√2√6 - 2√3√7
= 3√2 + 3√12 - 2√21
= 3√2 + 3√(2*2*3) - 2√21
= 3√2 + 6√3 - 2√21
b. (2 + √5)(2-√5)
= 2*2 + 2*(-√5) + 2√5 + (√5)(-√5)
= 4 - 2√5 + 2√5 - 5
= -1
c. 2³√3 x ³√72 - ⁴√125 x ⁴√5
= (2³√3 x ³√(2*2*2*3) - ⁴√(5*5*5) x ⁴√5
= 2³√3 x ³√(2³√3) - ⁴√(5⁴√5)
= 2³√3 x 2√3 - 5√5
= 2^(3+1)√(3*3) - 5√5
= 2^4 * 3 - 5√5
= 16√3 - 5√5
d. (4√27 + 3√2)(4√3 - √2)
= (4√(3*3*3) + 3√2)(4√3 - √2)
= 4√3 * 3 + 3√2 * 4√3 - 4√3 * √2 - 3√2 * √2
= 12√3 + 12√(2*3) - 4√(3*2) - 3√(2*2)
= 12√3 + 12√6 - 4√6 - 3√4
= 12√3 + 12√6 - 4√6 - 3 * 2
= 12√3 + 12√6 - 4√6 - 6
= 12√3 + 8√6 - 6