SURDS AND INDICES

1. Explanation :

$$1024^{-4/5}$$ = $$(4^5)^{-4/5}$$ =$$4 ^{ 5 * ( (-4) / 5 )}$$ = $$4^{-4}$$= 1 / $$4^4$$ = 1 / 256

1. Explanation :

1. Explanation :

1. Explanation :

$$2^{x - 1}$$ + $$2^{x+ 1}$$ = 1280
2x-1 (1 +$$2^2$$)= 1280
$$2^{x-1}$$ = 1280 / 5 = 256 = $$2^8$$
x -1 = 8
x = 9.

1. Explanation :

1. Explanation :

$$(16)^{3/2}$$ + $$(16)^{-3/2}$$= $$(4^2)^{3/2}$$ +$$(4^2)^{-3/2}$$= $$4^{2 * 3/2}$$ +$$4^{ 2* (-3/2)}$$=
= $$4^3$$ +$$4^{-3}$$ = $$4^3$$ + (1/$$4^3$$ = ( 64 + ( 1/64)) = 4097/64.

1. Explanation :

$$(1/5)^3y$$ = 0.008 = 8/1000 = 1/125 = $$(1/5)^3$$
3y = 3
Y = 1.
$$(0.25)^y$$ = $$(0.25)^1$$ = 0.25.

1. Explanation :

1. Explanation :

1. Explanation :

$$z^1$$= $$x^c$$ =$$(y^a)^c$$ [since x= ya]
=$$y^{(ac)}$$ = $$(zb)^{ac}$$ [since y=zb]
=$$zb^{(ac)}$$= $$z^{abc}$$
abc = 1.

1. Explanation :

Given surds are of order 2 and 3. Their L.C.M. is 6. Changing each to a surd of order 6, we get:
$$\sqrt { 2}$$ = $$2^{1/2}$$= $$2^{(1/2)*(3/2)}$$=$$2^{3/6}$$=$$8^{1/6}$$=$$\sqrt [6]{8}$$
$$\sqrt [3]{3}$$= $$3^{1/3}$$=$$3^{(1/3)*(2/2)}$$==$$3^{2/6}$$ = $$(3^2)^{1/6}$$ = $$(9)^{1/6}$$ = $$\sqrt[6]{9}$$
Clearly, 6√9 > 6√8 and hence 3√3 > √2.

1. Explanation :

Given surds are of order 4, 2 and 3 respectively. Their L.C,M, is 12, Changing each to a surd of order 12, we get: