If the rate of inflation for the next 20 years is 2.5% per year, what annual income will be needed 20 years from now to have the same purchasing power as a $30,000 annual income today?

Correct Answer: a

Explanation:

Given, xyz = 1, a^x = b, b^y = c
Now, b = a^x
Apply the law of algebra .
Take the power y of both side equation.
? b^y= a^xy
Take the power z of both side equation.
? b^yz = a^xyz
? ( b^y )^z = a^xyz
Replace the b^y with c and xyz with 1 as per given equation,
? c^z = a

Correct Answer: 2

Explanation:

Here, Let us Assume ,
p = ?56 + ?56 + ?56 + ...........
? p = ?56 + p
Square on both side and solve the above equation.
? p²= 56 + p
? p² - p= 56
? p(p - 1 ) = 56
? p(p - 1 ) = 8 x 7
? Either p = 8 or p - 1 = 7
? So p = 8;
Now given equation in question is
?56 + ?56 + ?56 + .... ) ÷ 2² = ?
? p ÷ 2² = ?
Put the value of p
? 8 ÷ 2² = ?
? 8 ÷ 4 = ?
? ? = 2;

Correct Answer: 1

Explanation:

( 9ⁿ x 3² x (3^-n/2)^-2 - (27)ⁿ ) / ( 3^3m x 2³ ) = 1/27
? ( 3²ⁿ x 3² x 3ⁿ - 3³ⁿ) / ( 3^3m x 2³ ) = 1/27
? ( 3^{2n + n} x 3² - 3³ⁿ) / ( 3^3m x 2³ )= 1/27
? ( 3³ⁿ x 3² - 3³ⁿ) / ( 3^3m x 2³ ) = 1/27
? 3³ⁿ( 3² - 1) / ( 3^3m x 8 ) = 1/27
? 3³ⁿ( 9 - 1) / (3^3m x 8 ) = 1/27
? ( 3³ⁿ x 8 ) / (3^3m x 8 ) = 1/27
? 3³ⁿ / 3^3m = 1/27
? 3^{3n - 3m} = 1/3³
? 3^{3( n - m )} = 3^-3
3(n - m) = -3
or n - m = -1
or m - n = 1

Correct Answer: 10

Explanation:

Given that , 289 = 17^x/5
Use the formula if a^x = a ^y then x = y will be true.
? 17^1/2 = 17^x/5
? x /5 = 1/2
? x = 10 .

Correct Answer: 3, 2

Explanation:

Apply the law of Fractional Exponents and Laws of Exponents
(a^m) x (aⁿ) = a^{m + n}
a^-m = 1/a^m
p⁰ = 1
and if p^X = p^Y then X will be equal to Y. means X = Y;
(a^m)ⁿ = a^{m x n}
(a^m) x (aⁿ) = a^{m + n}
a^m ÷ aⁿ = a^{m ? n}
Or
a^m/aⁿ = a^{m ? n}

Given equation is
2^p + 3^q = 17 ----------------(i)
2^{p + 2} - 3^{q + 1} = 5
? 2^p x 2² - 3^q x 3 = 5
? 2^p x 4 - 3q x 3 = 5
? 4 x 2^p - 3 x 3q = 5---------------(ii)
On multiplying Eq. (i) by 3 and adding it, with Eq. (ii), we get

3 x 2^p + 3 x 3^q = 51
4 x 2^p - 3 x 3^q = 5
_____________________________
7 x 2^p = 56

? 7 x 2^p = 7 x 8
? 2^p = 8
? 2^p = 2³
? p = 3

On substituting the value of p in Eq. (i) . we get
2³ + 3^q = 17
? 8 + 3^q = 17
? 3^q = 17 - 8
? 3^q = 9
? 3^q = 3²
? q = 2

Correct Answer: 458

Explanation:

Given that, (42 x 229) ÷ (9261)^1/3 = ?
As we know that (9621)^1/3 = (21 x 21 x 21)^1/3 = 21^3x1/3
? 42 x 229 ÷ (21^{3 x} )^1/3 = ?
? 42 x 229 ÷ 21 = ?
? 42 x 229 x 1/ 21 = ?
? 2 x 229 = ?
? ? = 2 x 229
? ? = 458

Correct Answer: 64

Explanation:

?5 + ?x = 3
Apply the law of Algebra Law.
Square both side of equation.
(?5 + ?x)² = 3²
? 5 + ?x = 9
? ?x = 9 - 5
? ?x = 4
Now cube the both side of equation,
? ( ?x ) ³ = 4³
? x = (4)³
? x = 64

Correct Answer: a = 11, b = -6

Explanation:

Given equation is,
( 5 + 2?3 ) / (7 + 4?3 )
Now Multiply and divide by ( 7 - 4?3 ) in above given equation.
By Rationalization, we will get,
[ ( 5 + 2?3 ) / ( 7 + 4?3 ) ] x [ ( 7 - 4?3 ) / ( 7 - 4?3 ) ]
[ ( 5 + 2?3 ) x ( 7 - 4?3 ) ] / [ ( 7 + 4?3 ) x ( 7 - 4?3 ) ]
Apply the formula and multiplication rule of algebra,
( P + Q ) x (R - S ) = P x R - P x S + Q x R - Q x S .................(1)
( a + b ) x ( a - b) = a² - b².......................................................(2)
= ( 5 x 7 - 5 x 4?3 + 7 x 2?3 - 2?3 x 4?3 )/ ( 7² - (4?3)² )
= 35 - 20?3 + 14?3 - 8 x 3 / ( 49 - 16 x 3 )
= ( 35 - 24 - 6?3 )/(49 - 48 )
= 11 - 6?3
? ( 5 + 2?3 ) /( 7 + 4?3 ) = a + b?3
? a + b?3 = 11 - 6?3
on equating the above equation and we get,
or a = 11 and b = - 6

Correct Answer: 125

Explanation:

Let the required number be N.
Then, (16/100) x (40/100) x N = 8
Then, N = (8 x 100 x 100) / (16 x 40)
= 125

Correct Answer: 1/250

Explanation:

2/5% = (2/5) /100 = 1/250