Question

Simplify the following surds : 1) 2√21 × √27 ÷ √343 2) 7√5 × √125 ÷ 2√27

Answer 1

Answer:

1) $\frac{18}{7}$

2) $\frac{175\sqrt{3}}{18}$

Step-by-step explanation:

* Lets explain how to simplify a square root

1)

∵ $2\sqrt{21}$ × $\sqrt{27}$ ÷ $\sqrt{343}$

∵ $\sqrt{21}=\sqrt{3}$ × $\sqrt{7}$

∴ $2\sqrt{21}$ = $2\sqrt{3}$ × $\sqrt{7}$

∵ $\sqrt{27}$ = $\sqrt{3}$ × $\sqrt{3}$ × $\sqrt{3}$

∵ $\sqrt{3}$ × $\sqrt{3}$ = 3

∴ $\sqrt{27}$ = $3\sqrt{3}$

∴ $2\sqrt{21}$ × $\sqrt{27}$ =

  $2\sqrt{3}$ × $\sqrt{7}$ × $3\sqrt{3}$

∵ $\sqrt{3}$ × $\sqrt{3}$ = 3

∵ 2 × 3 × 3 = 18

∴ $2\sqrt{21}$ × $\sqrt{27}$ = $18\sqrt{7}$

∵ $\sqrt{343}$ = $\sqrt{7}$ × $\sqrt{7}$ × $\sqrt{7}$

∵ $\sqrt{7}$ × $\sqrt{7}$ = 7

∴ $\sqrt{343}$ = $7\sqrt{7}$

∵ $2\sqrt{21}$ × $\sqrt{27}$ ÷ $\sqrt{343}$ =

  $18\sqrt{7}$ ÷ $7\sqrt{7}$

∵ $\sqrt{7}$ ÷ $\sqrt{7}$ = 1

∴ $2\sqrt{21}$ × $\sqrt{27}$ ÷ $\sqrt{343}$ =

  $\frac{18}{7}$

2)

∵ $7\sqrt{5}$ × $\sqrt{125}$ ÷ $2\sqrt{27}$  

∵ $\sqrt{125}$ = $\sqrt{5}$ × $\sqrt{5}$ × $\sqrt{5}$

∵ $\sqrt{5}$ × $\sqrt{5}$ = 5

∴ $\sqrt{125}$ = $5\sqrt{5}$

∴ $7\sqrt{5}$ × $\sqrt{125}$ =

  $7\sqrt{5}$ × $5\sqrt{5}$

∵ $\sqrt{5}$ × $\sqrt{5}$ = 5

∴ $7\sqrt{5}$ × $\sqrt{125}$ = 7 × 5 × 5 = 175

∵ $2\sqrt{27}$ = $2\sqrt{3}$ × $\sqrt{3}$ × $\sqrt{3}$

∵ $\sqrt{3}$ × $\sqrt{3}$ = 3

∴ $2\sqrt{27}$ = $6\sqrt{3}$

∴ $7\sqrt{5}$ × $\sqrt{125}$ ÷ $2\sqrt{27}$ =

  175 ÷ $6\sqrt{3}$ = $\frac{175}{6\sqrt{3}}$

∵ $\frac{175}{6\sqrt{3}}$ not in the simplest form because

  the denominator has square root

∴ Multiply up and down by $\sqrt{3}$

∴  $\frac{175}{6\sqrt{3}}$ = $\frac{175\sqrt{3}}{6\sqrt{3}*\sqrt{3}}$

∴  $\frac{175}{6\sqrt{3}}$ = $\frac{175\sqrt{3}}{18}$

∴ $7\sqrt{5}$ × $\sqrt{125}$ ÷ $2\sqrt{27}$ =

  $\frac{175\sqrt{3}}{18}$