Simplify the following:
(i) √45 – 3√20 + 4√5
(ii) 3√3 + 2√27 + 7/√3
(iii) 6√5 × 2√5
(iv) 8√15 ÷ 2√3
(v) √24/8 + √54/9
(vi) 3/√8 + 1/√2
Solution:
(i)√45 – 3√20 + 4√5
Let us simplify the expression,
√45 – 3√20 + 4√5
= √(9×5) – 3√(4×5) + 4√5
= 3√5 – 3×2√5 + 4√5
= 3√5 – 6√5 + 4√5
= √5
(ii)3√3 + 2√27 + 7/√3
Let us simplify the expression,
3√3 + 2√27 + 7/√3
= 3√3 + 2√(9×3) + 7√3/(√3×√3) (by rationalizing)
= 3√3 + (2×3)√3 + 7√3/3
= 3√3 + 6√3 + (7/3) √3
= √3 (3 + 6 + 7/3)
= √3 (9 + 7/3)
= √3 (27+7)/3
= 34/3 √3
(iii) 6√5 × 2√5
Let us simplify the expression,
6√5 × 2√5
= 12 × 5
= 60
(iv)8√15 ÷ 2√3
Let us simplify the expression,
8√15 ÷ 2√3
= (8 √5 √3) / 2√3
= 4√5
(v)√24/8 + √54/9
Let us simplify the expression,
√24/8 + √54/9
= √(4×6)/8 + √(9×6)/9
= 2√6/8 + 3√6/9
= √6/4 + √6/3
By taking LCM
= (3√6 + 4√6)/12
= 7√6/12
(vi)3/√8 + 1/√2
Let us simplify the expression,
3/√8 + 1/√2
= 3/2√2 + 1/√2
By taking LCM
= (3 + 2)/(2√2)
= 5/(2√2)
By rationalizing,
= 5√2/(2√2 × 2√2)
= 5√2/(2×2)
= 5√2/4
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