Surds and Indices Quiz-1

Explanation:-

4^x=1024 ⇔ 4^x=4⁵
Then;
x=5
5²+3=28.

Explanation:-
\frac{1}{1+ a^{(n-m)}}\frac{1}{1+a^{(m-n)}} =
\frac{1}{1+\frac{a^{n}}{a^{m}}} + \frac{1}{1+\frac{a^{m}}{a^{n}}}
\frac{a^{m}}{a^{m}+a^{n}} + \frac{a^{n}}{a^{m}+a^{n}}
\frac{a^{m}+a^{n}}{a^{m}+a^{n}}=1.

Explanation:-
\left ( \sqrt{8} \right )^{1/3}=\left ( 8^{1/2} \right )^{1/3}=8^{1/6} = \left ( 2^{3} \right )^{1/6}= 2^{1/2}= \sqrt{2}

Explanation:-
We know that 12^{2}=144.
putting a= 12 and b=2 ,we get
\left ( a-1 \right )^{b+1} = \left ( 12-1 \right )^{2+1} = 11^{3} =1331.

Explanation:-
=\left ( 81 \right )^{0.16}\times \left ( 81 \right )^{0.09}<br /> <br /> = \left ( 81 \right )^{\left ( 0.16+0.009 \right )}
=\left ( 81 \right )^{0.25}
=\left ( 81 \right )^{\left ( 25/100 \right )}
=\left ( 3^{4} \right )^{\left ( 1/4 \right )}
=3

Explanation:-

4^{(a - b)} = 1024 = 4⁵ ⇔ a-b= 5……(i).
5^{(a + b)} = 3125 = 5⁵ ⇔ a+b= 5…(ii).

On solving (i) and (ii), we get a = 5

Explanation:-
Cube root of 1728 is 12. Therefore,
\left ( 12^{3} \right )^{-2/3}
12^{-2}=\frac{1}{12^{2}}=\frac{1}{144}.

Explanation:-
4³ =64, 4⁴=256,3⁶=729.
\frac{1}{4^{3^{-(2/3)}}}+\frac{1}{4^{4^{-(3/4)}}} +\frac{1}{3^{6^{-(3/6)}}}.
\frac{1}{(4)^{-2}} + \frac{1}{(4)^{-3}} +\frac{1}{(3)^{-3}}
4^{2}+4^{3}+3^{3} = 107.

Explanation:-
3^{x} \times 27^{(1/4)} =3^{(1/4)}.
3^{x} \times (3^{3})^{1/4} =3^{(1/4)}.
3^{x} \times 3^{(3/4)} =3^{(1/4)}
3^{[x+(3/4)]} =3^{(1/4)}.
x+ \frac{3}{4}=\frac{1}{4}.
x=\frac{1}{4}-\frac{3}{4}=\frac{-2}{4}=\frac{-1}{2}.

Explanation:-
5^x-2
=\frac{5^{x}}{5^{2}}
=\frac{1250}{25}=50.