Question

please answer this question​

Answer 1

5. Represent √7.5 on a number line. Also, justify your answer.

We know that,

∆ OBD is a right-angled triangle. [Refer attachment]

The radius of the circle AC is $\sf \dfrac{7.5 + 1}{2} units.$

Therefore, OC = OD = OA = $\sf \dfrac{7.5 +1}{2} units.$

Now, OB = $\sf OB = 7.5 - \bigg(\dfrac{7.5+1}{2} \bigg) = \dfrac{7.5 + 1}{2}$

By Pythagoras theorem:-

BD² = OD² - OB²

$\implies \sf \bigg(\dfrac{7.5 - 1}{2} \bigg)^{2} - \bigg(\dfrac{7.5 - 1}{2} \bigg) = \dfrac{4 (7.5)}{4}$

BD² = 7.5 units.

BD = √7.5 units.

6.

$\implies \sf \: \dfrac{ {(243)}^{ \frac{3}{5} } {(25)}^{ \frac{3}{2} }}{ {625}^{ \frac{1}{2} } \times {(8)}^{ \frac{4}{3} } \times {(16)}^{ \frac{5}{4} } }$

$\implies \sf \dfrac{ {(3)}^{ 5 \times \frac{3}{5} } \times \: {(5)}^{2 \times \frac{3}{2} } }{ {5}^{4 \times \frac{1}{2} } \times {(2)}^{3 \times \frac{4}{3} } \times {(2)}^{4 \times \frac{5}{4} } }$

$\implies \sf \: \dfrac{ {(5)}^{3 \times } {(3)}^{3} }{ {(5)}^{2} \times {(2)}^{5 + 4} }$

$\implies \sf \: \dfrac{125 \times 27}{25 \times 512}$

$\implies \sf \boxed{ \dfrac{3375}{12800} }$