Solution:

$$\frac{2}{3√5}$$ × $$\frac{√5}{√5}$$

= $$\frac{2√5}{3×5}$$

= $$\frac{2√5}{15}$$

Solution:

$$\frac{3}{√2}$$×$$\frac{√2}{√2}$$

=$$\frac{3√2}{√2^2}$$

=$$\frac{3√2}{2}$$

Solution:

$$\frac{5+√3}{√5}$$×$$\frac{√5}{√5}$$

=$$\frac{5√5+√15}{5}$$

Solution:

3√5+6√5

=(3+6)√5

=9√5

Solution:

3√10 - 3√10

= (3-3)√10

= 0×√10

= 0

Solution:

3√20+2√45

= 3$$\sqrt{2×2×5}$$+2$$\sqrt{3×3×5}$$

= 3$$\sqrt{2^2×5}$$+2$$\sqrt{3^2×5}$$

= 3×2√5+2×3√5

= 6√5+6√5

= (6+6)√5

= 12√5

Solution:

(5√7 × 3√5) × 4√3

= 15$$\sqrt{7×5}$$ × 4√3

= 15√35 × 4√3

= 60$$\sqrt{35×3}$$

= 60√105

Solution:

(2√3 × 3√5) + 5√15

= (6$$\sqrt{3×5}$$ + 5√15

= (6$$\sqrt{15}$$ + 5√15

= (6+5)√15

=11√15

Solution:

√125-√45

= $$\sqrt{25×5}$$ - $$\sqrt{9×5}$$

= 5√5 - 3√5

= 2√5

Solution:

3√2 - 4√2 + 5√2

= 3$$\sqrt{2}$$ + 5$$\sqrt{2}$$ - 4√2

= 8√2 - 4√2

= 4√2

Solution:

$$\sqrt{128}$$ - $$\sqrt{50}$$

= $$\sqrt{2\times2\times2\times2\times2\times2\times2}$$ - $$\sqrt{2\times5\times5}$$

= $$\sqrt{2^2\times2^2\times2^2\times2}$$ - $$\sqrt{2\times5^2}$$

= 2$$\times$$2$$\times$$2$$\sqrt{2}$$ - 5$$\sqrt{2}$$

= 8$$\sqrt{2}$$ - 5$$\sqrt{2}$$

= (8 - 5) $$\sqrt{2}$$

= 3$$\sqrt{2}$$

Solution:

$$\frac{√6+√10}{√2}$$

=$$\frac{√6+√10}{√2}$$×$$\frac{√2}{√2}$$

= $$\frac{√2(√6+√10)}{2}$$

= $$\frac{(√2+√20)}{√2}$$

= $$\frac{√4×3+√4×5}{2}$$

= $$\frac{2√3+2√5}{2}$$

= $$\frac{2(√3+√5)}{2}$$

= √3+√5

Soutioln:

√288 - √72 + √8

= $$\sqrt{2×2×2×2×2×3×3}$$ - $$\sqrt{2×2×2×3×3}$$ + $$\sqrt{2×2×2}$$

= $$\sqrt{2^2 × 2^2 × 2 × 3^2}$$ - $$\sqrt{2^2× 2 × 3^2}$$ + $$\sqrt{2^2 ×2}$$

= 2 × 2 × 3√2 - 2 × 3√2 + 2√2

= (12-6+2)√2

= (14-6)√2

= 8√2

Solution:

√128-√50

= $$\sqrt{2×2×2×2×2×2×2}$$-$$\sqrt{2×5^2}$$

=$$\sqrt{2^2×2^2×2^2×2}$$-$$\sqrt{2×5^2}$$

= 2 × 2 × 2√2 - 5√2

= 8√2 - 5√2

= (8-5)√2

= 3√2

Solution:

$$\frac{3}{√2}$$+ 5

= $$\frac{3×√2}{√2×√2}$$+ 5

= $$\frac{3√2}{(√2)^2}$$+ 5

= $$\frac{3√2}{2}$$+ 5

= $$\frac{3×√2+10}{2}$$

Solution:

$$\frac{2}{√5}$$+$$\frac{3}{√2}$$

=$$\frac{2}{√5}$$×$$\frac{√5}{√5}$$+$$\frac{3}{√2}$$+$$\frac{√2}{√2}$$ (Rationalise the denominator of each term)

=$$\frac{2√5}{5}$$+$$\frac{3√2}{2}$$

=$$\frac{4√5+15√2}{10}$$ (LCM of 5 and 2 = 10)

Solution:

$$\sqrt{63}$$ - 2$$\sqrt{28}$$ + 5 $$\sqrt{7}$$

= $$\sqrt{3\times3\times7}$$ - 2$$\sqrt{2\times2\times7}$$ + 5$$\sqrt{7}$$

=$$\sqrt{3^2\times7}$$ - 2$$\sqrt{2^2\times7}$$ + 5$$\sqrt{7}$$

= 3$$\sqrt{7}$$ - 2$$\times$$2$$\sqrt{7}$$ + 5$$\sqrt{7}$$

= 3 $$\sqrt{7}$$ - 4$$\sqrt{7}$$ + 5$$\sqrt{7}$$

= (3-4+5)$$\sqrt{7}$$

= (8-4)$$\sqrt{7}$$

= 4$$\sqrt{7}$$

Solution:

(3 $$\sqrt{7}$$ + 2$$\sqrt{28}$$ $$\times$$ 4$$\sqrt{7}$$

=( 3 $$\sqrt{7}$$ + 2 $$\sqrt{2\times2\times7}$$) $$\times$$ 4$$\sqrt{7}$$

= (3 $$\sqrt{7}$$ + 2$$\times$$2$$\sqrt{7}$$ $$\times$$ 4$$\sqrt{7}$$

= (3$$\sqrt{7}$$ + 4$$\sqrt{7}$$ ) $$\times$$ 4$$\sqrt{7}$$

= (3+4) $$\sqrt{7}$$ $$\times$$ 4$$\sqrt{7}$$

= 7 $$\sqrt{7}$$ $$\times$$ 4$$\sqrt{7}$$

= 28$$\sqrt{7\times7}$$

= 28$$\sqrt{7^2}$$

= 28$$\times$$7

= 196

Solution:

21$$\sqrt{7}$$ - 3$$\sqrt{28}$$ + $$\sqrt{63}$$

= 21$$\sqrt{7}$$ - 3$$\sqrt{2\times2\times7}$$ + $$\sqrt{3\times3\times7}$$

= 21$$\sqrt{7}$$ - 3$$\sqrt{2^2\times7}$$ + $$\sqrt{3^2\times7}$$

= 21$$\sqrt{7}$$ - 3$$\times$$2$$\sqrt{7}$$ + 3$$\sqrt{7}$$

= 21$$\sqrt{7}$$ - 6$$\sqrt{7}$$ + 3$$\sqrt{7}$$

= (21 - 6 + 3)$$\sqrt{7}$$

= 18$$\sqrt{7}$$

Solution:

3$$\sqrt{20}$$ + 2 $$\sqrt{45}$$

= 3$$\sqrt{2\times2\times5}$$ + 2$$\sqrt{3\times3\times5}$$

= 3$$\times$$2 $$\sqrt{5}$$ + 2$$\times$$3$$\sqrt{5}$$

= 3$$\sqrt{2^2\times5}$$ + 2$$\sqrt{3^2\times5}$$

= 3$$\times$$2$$\sqrt{5}$$ + 2$$\times$$3$$\sqrt{5}$$

= 6$$\sqrt{5}$$ + 6$$\sqrt{5}$$

= (6+6)$$\sqrt{5}$$

= 12$$\sqrt{5}$$