R.D. Sharma solutions for माठेमटिक्स १ [इंग्रजी] इयत्ता ९ chapter 2 - Exponents of Real Numbers [Latest edition]

Simplify the following

`3(a^4b^3)^10xx5(a^2b^2)^3`

Simplify the following

`(2x^-2y^3)^3`

Simplify the following

`((4xx10^7)(6xx10^-5))/(8xx10^4)`

Simplify the following

`(4ab^2(-5ab^3))/(10a^2b^2)`

Simplify the following

`((x^2y^2)/(a^2b^3))^n`

Simplify the following

`(a^(3n-9))^6/(a^(2n-4))`

If a = 3 and b = -2, find the values of :

a^a + b^b

If a = 3 and b = -2, find the values of :

a^b + b^a

If a = 3 and b = -2, find the values of :

(a + b)^ab

Prove that:

`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)=1`

Prove that:

`(x^a/x^b)^cxx(x^b/x^c)^axx(x^c/x^a)^b=1`

Prove that:

`1/(1+x^(a-b))+1/(1+x^(b-a))=1`

Prove that:

`1/(1 + x^(b - a) + x^(c - a)) + 1/(1 + x^(a - b) + x^(c - b)) + 1/(1 + x^(b - c) + x^(a - c)) = 1`

Prove that:

`(a+b+c)/(a^-1b^-1+b^-1c^-1+c^-1a^-1)=abc`

Prove that:

`(a^-1+b^-1)^-1=(ab)/(a+b)`

If abc = 1, show that `1/(1+a+b^-1)+1/(1+b+c^-1)+1/(1+c+a^-1)=1`

Simplify the following:

`(3^nxx9^(n+1))/(3^(n-1)xx9^(n-1))`

Simplify the following:

`(5xx25^(n+1)-25xx5^(2n))/(5xx5^(2n+3)-25^(n+1))`

Simplify the following:

`(5^(n+3)-6xx5^(n+1))/(9xx5^x-2^2xx5^n)`

Simplify the following:

`(6(8)^(n+1)+16(2)^(3n-2))/(10(2)^(3n+1)-7(8)^n)`

Solve the following equation for x:

`7^(2x+3)=1`

Solve the following equation for x:

`2^(x+1)=4^(x-3)`

Solve the following equation for x:

`2^(5x+3)=8^(x+3)`

Solve the following equation for x:

`4^(2x)=1/32`

Solve the following equation for x:

`4^(x-1)xx(0.5)^(3-2x)=(1/8)^x`

Solve the following equation for x:

`2^(3x-7)=256`

Solve the following equations for x:

`2^(2x)-2^(x+3)+2^4=0`

Solve the following equations for x:

`3^(2x+4)+1=2.3^(x+2)`

If 49392 = a⁴b²c³, find the values of a, b and c, where a, b and c are different positive primes.

If `1176=2^a3^b7^c,` find a, b and c.

Given `4725=3^a5^b7^c,` find

(i) the integral values of a, b and c

(ii) the value of `2^-a3^b7^c`

If `a=xy^(p-1), b=xy^(q-1)` and `c=xy^(r-1),` prove that `a^(q-r)b^(r-p)c^(p-q)=1`

Assuming that x, y, z are positive real numbers, simplify the following:

`(sqrt(x^-3))^5`

Assuming that x, y, z are positive real numbers, simplify the following:

`sqrt(x^3y^-2)`

Assuming that x, y, z are positive real numbers, simplify the following:

`(x^((-2)/3)y^((-1)/2))^2`

Assuming that x, y, z are positive real numbers, simplify the following:

`(sqrtx)^((-2)/3)sqrt(y^4)divsqrt(xy^((-1)/2))`

Assuming that x, y, z are positive real numbers, simplify the following:

`root5(243x^10y^5z^10)`

Assuming that x, y, z are positive real numbers, simplify the following:

`(x^-4/y^-10)^(5/4)`

Assuming that x, y, z are positive real numbers, simplify the following:

`(sqrt2/sqrt3)^5(6/7)^2`

Simplify:

`(16^(-1/5))^(5/2)`

Simplify:

`root5((32)^-3)`

Simplify:

`root3((343)^-2)`

Simplify:

`(0.001)^(1/3)`

Simplify:

`((25)^(3/2)xx(243)^(3/5))/((16)^(5/4)xx(8)^(4/3))`

Simplify:

`(sqrt2/5)^8div(sqrt2/5)^13`

Simplify:

`((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)`

Prove that:

`sqrt(3xx5^-3)divroot3(3^-1)sqrt5xxroot6(3xx5^6)=3/5`

Prove that:

`9^(3/2)-3xx5^0-(1/81)^(-1/2)=15`

Prove that:

`(1/4)^-2-3xx8^(2/3)xx4^0+(9/16)^(-1/2)=16/3`

Prove that:

`(2^(1/2)xx3^(1/3)xx4^(1/4))/(10^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(4^(-3/5)xx6)=10`

Prove that:

`sqrt(1/4)+(0.01)^(-1/2)-(27)^(2/3)=3/2`

Prove that:

`(2^n+2^(n-1))/(2^(n+1)-2^n)=3/2`

Prove that:

`(64/125)^(-2/3)+1/(256/625)^(1/4)+(sqrt25/root3 64)=65/16`

Prove that:

`(3^-3xx6^2xxsqrt98)/(5^2xxroot3(1/25)xx(15)^(-4/3)xx3^(1/3))=28sqrt2`

Prove that:

`((0.6)^0-(0.1)^-1)/((3/8)^-1(3/2)^3+((-1)/3)^-1)=(-3)/2`

Show that:

`1/(1+x^(a-b))+1/(1+x^(b-a))=1`

Show that:

`[{x^(a(a-b))/x^(a(a+b))}div{x^(b(b-a))/x^(b(b+a))}]^(a+b)=1`

Show that:

`(x^(1/(a-b)))^(1/(a-c))(x^(1/(b-c)))^(1/(b-a))(x^(1/(c-a)))^(1/(c-b))=1`

Show that:

`(x^(a^2+b^2)/x^(ab))^(a+b)(x^(b^2+c^2)/x^(bc))^(b+c)(x^(c^2+a^2)/x^(ac))^(a+c)=x^(2(a^3+b^3+c^3))`

Show that:

`(x^(a-b))^(a+b)(x^(b-c))^(b+c)(x^(c-a))^(c+a)=1`

Show that:

`{(x^(a-a^-1))^(1/(a-1))}^(a/(a+1))=x`

Show that:

`(a^(x+1)/a^(y+1))^(x+y)(a^(y+2)/a^(z+2))^(y+z)(a^(z+3)/a^(x+3))^(z+x)=1`

Show that:

`(3^a/3^b)^(a+b)(3^b/3^c)^(b+c)(3^c/3^a)^(c+a)=1`

If 2^x = 3^y = 12^z, show that `1/z=1/y+2/x`

If 2^x = 3^y = 6^-z, show that `1/x+1/y+1/z=0`

If a^x = b^y = c^z and b² = ac, show that `y=(2zx)/(z+x)`

If 3^x = 5^y = (75)^z, show that `z=(xy)/(2x+y)`

If `27^x=9/3^x,` find x.

Find the value of x in the following:

`2^(5x)div2x=root5(2^20)`

Find the value of x in the following:

`(2^3)^4=(2^2)^x`

Find the value of x in the following:

`(3/5)^x(5/3)^(2x)=125/27`

Find the value of x in the following:

`5^(x-2)xx3^(2x-3)=135`

Find the value of x in the following:

`2^(x-7)xx5^(x-4)=1250`

Find the value of x in the following:

`(root3 4)^(2x+1/2)=1/32`

Find the value of x in the following:

`5^(2x+3)=1`

Find the value of x in the following:

`(13)^(sqrtx)=4^4-3^4-6`

Find the value of x in the following:

`(sqrt(3/5))^(x+1)=125/27`

If `x=2^(1/3)+2^(2/3),` Show that x³ - 6x = 6

Determine `(8x)^x,`If `9^(x+2)=240+9^x`

If `3^(x+1)=9^(x-2),` find the value of `2^(1+x)`

If `3^(4x) = (81)^-1` and `10^(1/y)=0.0001,` find the value of `  2^(-x+4y)`.

If `5^(3x)=125` and `10^y=0.001,` find x and y.

Solve the following equation:

`3^(x+1)=27xx3^4`

Solve the following equation:

`4^(2x)=(root3 16)^(-6/y)=(sqrt8)^2`

Solve the following equation:

`3^(x-1)xx5^(2y-3)=225`

Solve the following equation:

`8^(x+1)=16^(y+2)` and, `(1/2)^(3+x)=(1/4)^(3y)`

Solve the following equation:

`4^(x-1)xx(0.5)^(3-2x)=(1/8)^x`

Solve the following equation:

`sqrt(a/b)=(b/a)^(1-2x),` where a and b are distinct primes.

If a and b are distinct primes such that `root3 (a^6b^-4)=a^xb^(2y),` find x and y.

If a and b are different positive primes such that

`((a^-1b^2)/(a^2b^-4))^7div((a^3b^-5)/(a^-2b^3))=a^xb^y,` find x and y.

If a and b are different positive primes such that

`(a+b)^-1(a^-1+b^-1)=a^xb^y,` find x + y + 2.

If `2^x xx3^yxx5^z=2160,` find x, y and z. Hence, compute the value of `3^x xx2^-yxx5^-z.`

If 1176 = `2^axx3^bxx7^c,` find the values of a, b and c. Hence, compute the value of `2^axx3^bxx7^-c` as a fraction.

Simplify:

`(x^(a+b)/x^c)^(a-b)(x^(b+c)/x^a)^(b-c)(x^(c+a)/x^b)^(c-a)`

Simplify:

`root(lm)(x^l/x^m)xxroot(mn)(x^m/x^n)xxroot(nl)(x^n/x^l)`

Show that:

`((a+1/b)^mxx(a-1/b)^n)/((b+1/a)^mxx(b-1/a)^n)=(a/b)^(m+n)`

If `a=x^(m+n)y^l, b=x^(n+l)y^m` and `c=x^(l+m)y^n,` Prove that `a^(m-n)b^(n-l)c^(l-m)=1`

If `x = a^(m + n), y = a^(n + l)` and `z = a^(l + m),` prove that `x^my^nz^l = x^ny^lz^m`

Write \[\left( 625 \right)^{- 1/4}\] in decimal form.

If 2⁴ × 4² =16^x, then find the value of x.

If 3^x-1 = 9 and 4^y+2 = 64, what is the value  of \[\frac{x}{y}\] ?

Write the value of  \[\sqrt[3]{7} \times \sqrt[3]{49} .\]

Write \[\left( \frac{1}{9} \right)^{- 1/2} \times (64 )^{- 1/3}\] as a rational number.

Write the value of \[\sqrt[3]{125 \times 27}\].

For any positive real number x, find the value of \[\left( \frac{x^a}{x^b} \right)^{a + b} \times \left( \frac{x^b}{x^c} \right)^{b + c} \times \left( \frac{x^c}{x^a} \right)^{c + a}\].

Write the value of \[\left\{ 5( 8^{1/3} + {27}^{1/3} )^3 \right\}^{1/4} . \]

Simplify \[\left[ \left\{ \left( 625 \right)^{- 1/2} \right\}^{- 1/4} \right]^2\]

For any positive real number x, write the value of  \[\left\{ \left( x^a \right)^b \right\}^\frac{1}{ab} \left\{ \left( x^b \right)^c \right\}^\frac{1}{bc} \left\{ \left( x^c \right)^a \right\}^\frac{1}{ca}\]

If (x − 1)³ = 8, What is the value of (x + 1)² ?

The value of \[\left\{ 2 - 3 (2 - 3 )^3 \right\}^3\] is 

The value of x − y^x-y when x = 2 and y = −2 is

The product of the square root of x with the cube root of x is

The seventh root of x divided by the eighth root of x is

The square root of 64 divided by the cube root of 64 is

Which of the following is (are) not equal to \[\left\{ \left( \frac{5}{6} \right)^{1/5} \right\}^{- 1/6}\] ?

When simplified \[( x^{- 1} + y^{- 1} )^{- 1}\] is equal to

If \[8^{x + 1}\] = 64 , what is the value of \[3^{2x + 1}\] ?

If (2³)² = 4^x, then 3^x =

If x^{-2 =} 64, then x^1/3+x⁰ =

When simplified \[\left( - \frac{1}{27} \right)^{- 2/3}\] is 

Which one of the following is not equal to \[\left( \sqrt[3]{8} \right)^{- 1/2} ?\]

Which one of the following is not equal to \[\left( \frac{100}{9} \right)^{- 3/2}\]?

If a, b, c are positive real numbers, then  \[\sqrt{a^{- 1} b} \times \sqrt{b^{- 1} c} \times \sqrt{c^{- 1} a}\] is equal to

 `(2/3)^x (3/2)^(2x)=81/16 `then x = 

The value of \[\left\{ 8^{- 4/3} \div 2^{- 2} \right\}^{1/2}\] is

If a, b, c are positive real numbers, then  \[\sqrt[5]{3125 a^{10} b^5 c^{10}}\]  is equal to

If a, m, n are positive ingegers, then \[\left\{ \sqrt[m]{\sqrt[n]{a}} \right\}^{mn}\] is equal to

If x = 2 and y = 4, then \[\left( \frac{x}{y} \right)^{x - y} + \left( \frac{y}{x} \right)^{y - x} =\]

The value of m for which \[\left[ \left\{ \left( \frac{1}{7^2} \right)^{- 2} \right\}^{- 1/3} \right]^{1/4} = 7^m ,\] is

The value of \[\left\{ \left( 23 + 2^2 \right)^{2/3} + (140 - 19 )^{1/2} \right\}^2 ,\] is 

(256)^0.16 × (256)^0.09

If 10^2y = 25, then 10^-y equals

If 9^x+2 = 240 + 9^x, then x =

If x is a positive real number and x² = 2, then x³ =

If \[\frac{x}{x^{1 . 5}} = 8 x^{- 1}\] and x > 0, then x =

If g = `t^(2/3) + 4t^(-1/2)`, what is the value of g when t = 64?

If \[4x - 4 x^{- 1} = 24,\] then (2x)^x equals

When simplified \[(256) {}^{- ( 4^{- 3/2} )}\] is

If \[\frac{3^{2x - 8}}{225} = \frac{5^3}{5^x},\]  then x =

The value of 64^-1/3 (64^1/3-64^2/3), is

If \[\sqrt{5^n} = 125\] then  `5nsqrt64`=

If (16)^2x+3 =(64)^x+3, then 4^2x-2 =

If \[2^{- m} \times \frac{1}{2^m} = \frac{1}{4},\] then \[\frac{1}{14}\left\{ ( 4^m )^{1/2} + \left( \frac{1}{5^m} \right)^{- 1} \right\}\]  is equal to

If \[\frac{2^{m + n}}{2^{n - m}} = 16\], \[\frac{3^p}{3^n} = 81\] and \[a = 2^{1/10}\],than  \[\frac{a^{2m + n - p}}{( a^{m - 2n + 2p} )^{- 1}} =\]

If \[\frac{3^{5x} \times {81}^2 \times 6561}{3^{2x}} = 3^7\]  then x =

If o <y <x, which statement must be true?

If 10^x = 64, what is the value of \[{10}^\frac{x}{2} + 1 ?\]

\[\frac{5^{n + 2} - 6 \times 5^{n + 1}}{13 \times 5^n - 2 \times 5^{n + 1}}\]  is equal to

If \[\sqrt{2^n} = 1024,\] then \[{3^2}^\left( \frac{n}{4} - 4 \right) =\]