Exponents Of Real Numbers

(i) ( )⁵ = (1 / x^3/2) ⁵

= (1 / x ^{3/2 × 5}) = (1 / x ^15/2)

(ii) (³ / y²) = (x³ / y²) ^1/2

= x^{3 × 1/2} / y^{2 × 1/2}

= x^3/2 / y

(iii) 1 / (x^2/3 y^1/2)²

= 1 / (x^{2/3 × 2} y^{1/2 × 2})

= 1 / x^4/3 y

(iv) (x^1/2)^-2/3 (y)² / (xy^-1/2)^1/2

= x^-1/3y² / (x^1/2y^-1/2×1/2)

= (x^-5/6) (y^9/4)

= (y^9/4) / (x^5/6)

(v) (243x¹⁰ y⁵ z¹⁰) ^1/5

= (3⁵)^1/5 x²yz²

= 3x²yz²

(vi) (y¹⁰ / x⁴)^5/4

= y ^{10 × 5/4} / x^{4 × 5/4}

= y^25/2 / x⁵

(i)

We know for any non-zero number a,

(a^m)ⁿ = a^mn

So,

As we know 4² = 16

Therefore,

= 4^-1

As we know for any non-zero number a,

a^-1 = 1/a

So 4^-1 = 1/4

(ii) [(343)^-2]^1/3

We know for any non-zero number a,

(a^m)ⁿ = a^mn

So,

As we know 7³ = 343

Therefore,

= 7^-2

As we know for any non-zero number a,

a^-1 = 1/a

So 7^-2= 1/7²

= 1/49

(iii) ()^1/3 = (1 / 10³) ^1/3

= = 0.1

(iv)

We know 25 = 5²

243 = 3⁵

16 = 2⁴

8 = 2³

So,

We know for any non-zero number a,

(a^m)ⁿ = a^mn

So,

(v)

We know that for any non-zero number a,

a^m ÷ aⁿ = a^m-n

So,

As we know for any non-zero number a,

a^-1 = 1/a

(vi)
We know for any non-zero number a,

(a^m)ⁿ = a^mn

So,

We know for any non-zero number a,

a^m × aⁿ = a^m+n

= 25 × 7

= 175

(i) (3^1/2+1/6.5^{-3/2 +1}) / (3^-1/3.5^1/2)

₌₍₃^2/3.5^-1/2) / (3^-1/3.5^1/2)

₌₍₃^{2/3 + 1/3}) / (5^{1/2 +1/2})

_=3/5

_{(ii) (3}² )^3/2 -3.1 – (1/9²)^-1/2

_{= 3}³ -3 -9

_{=27 -3 -9}

_=27-12

₌₁₅

_{(iii) 2}^(-2)(-2) -3.8^2/3 +(3/4)^-1

₌₂⁴ -3.2² + 4/3

_{=16 -12 + 4/3}

_=16/3

_{(iv) [(2.3}^1/3)/(2^-1/5 5^2/5)] × (2^-1/5.3)/ (3^4/3.5^7/5)

_{= 2.3}^{1/3 +1 -4/3} / 5^2/5-7/5

_{= 2.5}

₌₁₀

_{(v) 1/2 + 1/(0.01)}^1/2 -3²

_{=1/2 + 10 – 9}

_{=1/2 + 1}

_=3/2

_{(vi) (2}ⁿ + 2^n-1)/ ) (2ⁿ⁺¹ - 2ⁿ)

₌₂ⁿ(1 + 2^-1 ) / 2ⁿ (2-1)

_{= [1 + (1/2)]/1}

_{=1 + 1/2}

_=3/2

_{(vii) (125/64)}^2/3 + (625/256)^1/4 + ( 5/4)

_=(5/4)² + 5/4 + 5/4

_{=25/16 + 5/4 + 5/4}

_=65/16

_{(viii) (3}^-3.6².7(2)^1/2)/ (5^4/3.(15)^-4/3‑.3^1/3) =28(2)^1/2

₍₃^-3.36.7(2)^1/2)/ (5^4/3-4/3.(3)^-1)

₍₃^-2.36.7(2)^1/2)/ (5⁰)

_1/9.36.7(2)^1/2

28

_{(ix) {1- 1/0.1}/ { (3/8)}^-1(3/2)³ + (-1/3)^-1

_{=1-10/{ (8/3)(3/2)}³ + (-3)

_=-9/(3²-3)

_{= -3/2}

We have,

(27)^x = 9 / 3^x

(3³)^x = 3² / 3^x

3^3x = 3 ^{2 – x}

3x = 2 – x {On equating exponents}

3x + x = 2

4x = 2

x = =

Hence, the value of x is

(i) we have,

2^5x÷ 2^x =

2^5x/2^x = 2^20/5

2^5x-x = 2⁴

4x = 4

x=1

(ii) We have,

(2³)⁴ = (2²)^x

2^3.4 = 2^2.x

12 = 2x

X=6

(iii) We have,

5^2x-x/ 3^2x-x = ()³

5^x/3^x = ()³

()^x = ()³

x=3

(iv) We have,

5^x-2 × 3^2x-3 = 135

5^x-2 × 3^2x-3 = 5

5^x-2 × 3^2x-3 = 5¹³

x-2 =1 ; 2x-3 =3

x=3 ; x= 3

(v) We have,

2^x-5 × 5^x-4 = 5

2^x-5 × 5^x-4 = 5¹ × 2⁰

x-5=0 ; x-4 = 1

x= 4 ; x=1 +4 =5

(vi) We have,

2^x-7 × 5^x-4 = 1250

2^x-7 × 5^x-4 = 2¹ × 5⁴

x – 7=1; x-4 = 4

x= 8; x= 4+4 = 8

{2-3(2-3)³}³

= {2 – 3 (-1)³}³

= {2 + 3}³

= 5³ = 125

= 0.2

The product law of exponent states that while multiplying two parts having same base, you can add the exponents.

(256)^0.16 × (256)^0.09

= (256) ^{0.16 + 0.09}

= (256) ^0.25

= 4^{4 ×} = 4

The quotient law of exponent states that to divide two exponents with the same base, you keep the base and subtract the powers.

10^2y = 25

_{= 10}^y = x

_{= x}² = 5²

_{= x = 5}

_{= 1/x = 10}^-y

_{= 1/5}

The power law of exponents states that:

(aⁿ)^m = a^n.m

Example: (2³)² = 2^3.2

= 2⁶ = 64

x – y ^{x – y}

= 2 – (-2)^{(2 + 2)}

= 2 – 16 = - 14

×

= x^1/2 × x^1/3

= x^5/6

x^{(a – b) (a + b)} × x^{(b – c) (b + c)} × x^{(c – a) (c + a)}

= x^{a.a - b.b} × x^{b.b – c.c} × x^{c.c – a.a}

= x^{a.a – b.b + b.b – c.c + c.c – a.a}

= x^o = 1

(x)^{ab × 1/ab} . (x)^{bc . 1/bc} . x^{ca . 1/ca}

_{= x . x . x}

= x³

9^{x + 2} = 240 + 9^x

9x × 9² = 240 + 9^x

Let 9^x = y

81y = 240 + y

80y = 240

y =

9^x = 3

3^2x = 3

2x = 1

x = = 0.5

{5(8^1/3 + 27^1/3)³}^1/4

= {5 (2 + 3)³}^1/4

= (5⁴)^1/4 = 5

x^1/7 / x^1/8

_{= (x)}^{1/7 – 1/8}

_{= (x)}^1/56

₌

As 64 can be written as 64 = 2×2×2×2×2×2
so 64 = 2⁶
√64=√(2⁶)=(2⁶)^1/2 = 2³=8

=(2⁶)^1/3 = 2²=4
= = 2

(x – 1)³ = 8

x – 1 = 2

x = 3

(x + 1)² = (3 + 1)²

= 4² = 16

_{= 1 / {(5/6)}^1/5}^1/5

_{= (5/6)}^-1/30

_{= (6/5)}^1/30

(x^-1 + y^-1)^-1

= ( + ) ^-1

= () ^-1

= ()

2⁴ × 2⁴ = 16x

2⁸ = 2⁴ × x

x = 2⁴ = 16

3^{x – 1} = 3²

_{x – 1 = 2}

_{x = 3}

₄^{y + 2} = 4³

_{y + 2 = 3}

_{y = 1}

_{x / y = 3/1 = 3}

8 ^{x + 1} – 64

= 8 ^{x + 1} = 8²

On equating powers, we get

x + 1 = 2

x = 1

= 3 ^{2x + 1}

= 3³ = 27

= (7 . 7²)^1/3

= (7)^3×1/3

= 7

Since, it is the property of square roots.

× (64)^-1/3

= (3²)^1/2 × (1/4³) ^-1/3

= 3 × 1/4 = 3/4

x² = 2

x =

x³ = (2)^{1/2 × 3}

= 2

= (5³ × 3³)^1/3

= 5 × 3

= 15

(2³)² = 2^2x

2x = 6

x = 3

can be written as: (10^x)^1/2 × 10

= (64)^1/2 × 10

= 8 × 10

= 80

g = t^2/3 + 4t^-1/2

= (64)^2/3 + 4 (64) ^-1/2

= [(64)^1/3]³ + 4 () ^1/2

= 4² + 4 ()

= 16 + =

()² = (8)²

= 8

x =

x^1/3 + x^o

= ()^1/3 + ()^o

= + 1 =

4^x – 4^{x – 1} = 24

Let 4x = y

y - = 24

4y – y = 96

y = 32

4^x = 32

2^2x = 2⁵

(2x)^x = (2 × )^5/2

= (5)^5/2 = 25

(-27) ^2/3

_{= (3)}³ × ^2/3

_{= 9}

1 / (8)^{-1/2 × 1/3}

_{= 2}^-1/2

₌

1 / (100/9)^3/2

_{= (10/3)}^{-3/2 × 2}

₌

1 / 256^1/8

_{= 1/2}^{8 × 1/8}

_{= 1/2}

5ⁿ (25 – 30) / 5ⁿ (13 – 10)

_{= -5 / 3}

(b/a)^1/2 × (c/b)^1/2 × (a/c)^1/2

_{= (b/a × c/b × a/c)}^1/2

_{= 1}

= 5^x × 3^{2x – 8} = 5⁵ × 3³

Comparing the coefficient of x we get,

= x = 5

(3/2)^-x (3/2)^2x = (3/2)⁴

= (3/2)^{-x + 2x} = (3/2)⁴

= -x + 2x = 4

= x = 4

= {2^-4 2^-2}^1/2

= {1/16 × 2²}^1/2

=

(3125a¹⁰b⁵c¹⁰)^1/5

= 5a²bc²

64^-1/3 (64^1/3 – 64^2/3)

= 4^-1 (4 – 4²)

= (4 – 16)

= = -3

= 125

5^n/2 = 5³

_{n/2 = 3}

_{n = 6}

5 = 5 (64)^1/6

= 5 (2)^6/6 = 10

4^{4x + 6} = 4 ^{3x + 9}

= 4x + 6 = 3x + 9

= x = 3

4^{2x – 2} = 4⁴

= 256

We know for any non-zero number a,

a^m × aⁿ = a^m+n

Again using (a^m)ⁿ = a^mn we get,

=a

2^-m × 1 / 2^m = 1/4

_{= 1/2}^m × 1/2^m = 1/4

_{= 1/4}^m = 1/4

_{= m = 1}

_{1/14 {(4}^m)^1/2 + (1/5^m)^-1}

_{= 1/14 {2 + 5}}

_{= 1/14 × 7}

₌

(2/4) ^{2 – 4} + (4 / 2)^{4 – 2}

= (1/2)^-2 + 2²

= 2² + 2²

= 8

[{7⁴}^-1/3]^1/4

= (1/7⁴)^{1/3 × 1/4}

= (1/7)^1/3 = 7^m

= 7 ^-1/3 = 7^m

= m = -1/3

2^{m + n - n + m} = 2⁴

2^2m = 2⁴

2m = 4

m = 2

Also a = 2^1/10




= a^{2m + m + n - 2n - p + 2p}

= a^{3m - n + p}

= {3² + 11}²

= (9 + 11)²

= (20)² = 400

= 2¹⁰

2^n/2 = 2¹⁰

= 10

= 3^{2 (n / 4 – 4)}

= 3^{2 (20/4 – 4)}

= 3² = 9

= 3⁷

= 3^5x × 3⁸ × 3⁸/ 3^2x = 3⁷

= 3^{5x + 16 – 2x} = 3⁷

= 5x + 16 – 2x = 7

= 3x + 16 =7

= 3x = -9

x = -3