## 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 (36

^{2})

^{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)}

= (4

^{3})

^{0.2}x (4

^{2})

^{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 }

(3

^{3})

^{ ⅓ = }(3)

^{1}

= 3

**Answer.**

## 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.

(16

^{2})

^{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.**

## If 5

Ques 6:= 3125, then the value of 5^{a}^{(a– 3)}is:

**Solution:**

According to the question:

5

*= 3125*

^{a}(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)

= 5

^{2}

= 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)}=3

^{3}

(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)}=3

^{5}

(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: