Tricks to Solve Surds and Indices Questions

Ques 1: 

Solve: 36^0.14 x  1296^0.18

Solution:
Given that a^b x a^c = a ^(b+c)
According to the question and as per the formula given above:
36^0.14 x 1296^0.18 (Make factors of 1296 = 36x36 i.e. equalizing the bases of the two figures)
36^0.14 x (36²)^0.18

36^0.14 x 36^0.36
36^{(0.14 + 0.36)}
36^0.5
36^(1/2)
i.e. Square root of 36 = √36 = √6x6
= 6 Answer.

Ques 2: 

Solve: 64^0.2 x 16^0.4 x 4^(-0.9)

Solution:
Given that a^b x a^c x a^d= a ^(b+c+d)
According to the question and as per the formula given above:
64^0.2 x 16^0.4 x 4^(-0.9)
= (4³)^0.2 x (4²)^0.4 x 4^(-0.9) (Make factors of 64 = 4x4x4 and factors of 16 = 4x4 i.e. equalizing the bases of the three figures)
4^{(3 x 0.2)} x 4^{(2 x 0.4)} x 4^(-0.9)
4^0.6 x 4^0.8 x 4^(-0.9)
4^{(0.6 + 0.8 - 0.9)}
4^{(1.4 - 0.9)}
4^0.5
= 4^½
= Square root of 4 = √4 = √2x2
= 2 Answer.

Ques 3: 

Solve: 27⅓

Solution:
Make three factors of 27 i.e. 3x3x3 so as to solve the denominator ^⅓
So 27^⅓ = (3x3x3) ^⅓
(3³) ^{⅓ =} (3)¹
= 3Answer.

Ques 4: 

Solve: (256) 0.16 x 16 0.18

Solution:
Given that a^b x a^c = a ^(b+c)
According to the question and as per the formula given above:
Make two factors of 256 = 16x16 so as to make the bases of the figures equal.
(16²)^0.16 x 16 ^0.18
= (16)^0.32 x 16 ^0.18
= (16)^0.5
= 16^½
= Square root of 16= √16= √4x4
= 4 Answer.

Ques. 5: 

Solve: If (a/b) ^x-1 = (b/a) ^x-3; find the value of x.

Solution:
According to the question:
(a/b) ^x-1 = (b/a) ^x-3
OR (a/b) ^x-1= (a/b) ^–(x-3)
(Reverse the inner equation so as to make both the equations similar and change the sign of + to – or vice versa)
OR (x-1) = -(x-3)
(When bases of both the equations are similar, then only the powers only would matter)
OR x -1 = - x + 3 (Solving the brackets)
OR x + x = 3+1
OR 2x = 4
X = 2 Answer.

Ques 6: 

If 5^a= 3125, then the value of 5^{(a– 3)} is:

Solution:
According to the question:
5^a = 3125
(Make factors of 3125 so as to make the bases of the two equations equal i.e. 3125 = 5 x 5 x 5 x 5 x 5)
Now (5) ^a = (5)⁵
(When bases of both the equations are similar, then only the powers only would matter)
a = 5
To find out is 5^{(a – 3)}
= 5^{(5- 3)} (Putting the value of a = 5 derived above)
= 5²
= 25 Answer

Ques 7: 

If 3(x– y)= 27 and 3(x + y) = 243, then find the value of x.

Solution:
According to the question:
3^(x – y) = 27 (Taking the first equation)
(Make factors of 27so as to make the bases of the two equations equal i.e. 27 = 3 x 3 x 3)
Thus: 3^(x – y) =3³
(When bases of both the equations are similar, then only the powers only would matter)
So x – y = 3
Now take the second part of the equation
3^(x + y) = 243
(Make factors of 243 so as to make the bases of the two equations equal i.e. 243= 3 x 3 x 3x3 x3)
Thus: 3^(x -+y) =3⁵
(When bases of both the equations are similar, then only the powers only would matter)
So x + y = 5
Now we have two linear equations i.e. x – y = 3and x + y = 5
Adding both these we get 2 x = 8
So x = 4 Answer.

Hereunder given are definitions and formulas:

A number that can’t be simplified into rational number by removal of a square root (or cube root etc) like √2 (square root of 2) can’t be simplified and thus it is a surd but √4 (square root of 4) can be simplified to 2, so it is NOT a surd.

Remember:

1. Surds cannot be simplified into rational numbers but indices can be simplified into rational numbers.
2. Any number raised to the power zero is always equals to one. (Eg: x 0= 1)
3. Every surd is an irrational number, but every irrational number is not a surd.
4. Important Formulas: