Nootan solutions for Mathematics [English] Class 9 ICSE chapter 6 - Indices/Exponents [Latest edition]

Simplify the following:

a⁷ × a⁵

Simplify the following:

`a^9/a^4`

Simplify the following:

`(a^7 xx a^4)/a^3`

Simplify the following:

`(x^4 xx x^5)/(x^3 xx x^-1)`

Simplify the following:

`(3x^4y^2(2xy)^5)/((2x^3y^3)^2`

Simplify the following:

`((x^(2a + 3))^5)/((x^(a + 1))^3`

Simplify the following:

`((xy^3)^2(3x^2y)^4)/(xy)^3`

Simplify the following:

`(x^3y^4z^2)/(x^2y^-1z^3)`

Simplify the following:

`(6^(2n + 6) - 6^3 * 36^(n + 1))/(6^(n + 2))^2`

Simplify the following:

`(2^(n + 2) xx 4^(n + 1))/(2^(n + 1) xx 4^(n - 2))`

Simplify the following:

`(5^(n + 5) - 6 xx 5^(n + 3))/(9 xx 5^(n + 1) - 20 xx 5^n)`

Simplify the following:

`(3^(n + 4) xx 3^((n - 2)(n + 2)))/(3^(n(n + 1)) xx 9^(n + 1))`

Write `(x^-1 - y^-1)/(x^-2 - y^-2)` in the simplest form.

Simplify:

`((a^(x + y))^(x - y)(a^(y - z))^(y + z))/((a^(x + z))^(x - z))`

Find the value of x and y from the following:

2^x · 3^y = 144

Find the value of x and y from the following:

3^x · 5^y = 675

Find the values of x, y and z from the following: 

2^x · 3^y · 5^z = 24000

If a = b^x, b = c^y and c = a^z prove that xyz = 1.

Prove that `(a^x/a^y)^(x + y) (a^y/a^z)^(y + z) (a^z/a^x)^(z + x) = 1`.

Prove that `(x^a/x^b)^(a^2 + ab + b^2) (x^b/x^c)^(b^2 + bc + c^2) (x^c/x^a)^(c^2 + ca + a^2) = 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 `(x + y + z)/((xy)^-1 + (yz)^-1 + (zx)^-1) = xyz`.

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

If `x = p^(m + n) * q^l, y = p^(n + l) * q^m, z = p^(l + m) * q^n` prove that `x^(m - l) * y^(n - l) * z^(l - m) = 1`.

If `a = x * y^(p - 1), b = x * y^(q - 1), c = x * y^(r - 1)`, prove that `a^(q - r) * b^(r - p) * c^(p - q) = 1`.

If 2^x = 3^y = 72^z, find the relation between x, y and z.

If 3^x = 5^y = 225^z, find the relation between x, y and z.

Evaluate the following:

`(243)^(1/5)`

Evaluate the following:

`(64)^(-3/2)`

Evaluate the following:

`(-32)^(2/5)`

Evaluate the following:

`(216)^(4/3)`

Evaluate the following:

`(25/9)^(3/2) + (27/64)^(-1/3)`

Evaluate the following:

`(81/256)^(1/4) xx (32/243)^((-2)/5)`

Evaluate the following:

`5^-1 xx 3^0 + 25^(1/2)`

Evaluate the following:

`3^-2 xx 2^0 - (27)^(-1/3)`

Evaluate the following:

`(27)^(4/3) - 5^0 xx (1/9)^(-3/2) + (81)^(1/2)`

Evaluate the following:

`(sqrt(3^4) xx root(3)(27))/(root(4)(81)) + (1/2)^-2`

Evaluate the following:

`(root(4)(2^8) xx root(5)(243) xx 5^0)/root(3)(216)`

If x = 2, y = 3, find x^x + y^–y.

If x = –2, y = –3, find x^x + y^y.

If x = 4, y = 3, find x^y + y^x.

If x = –1, y = 2, z = –3, find x^y + y^x + z^x.

If x = 2, y = 3, z = 4, find x^z + y^–x + z^y.

Solve the following equation:

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

Solve the following equation:

`3^(5x - 1) = root(4)(3)`

Solve the following equation:

`5^(3x - 1) = root(3)(5)`

Solve the following equation:

`2^(4x + 1) = root(4)(256)`

Solve the following equation:

`2^(3x + 1) = 16 xx 2^(2x)`

Solve the following equation:

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

Solve the following equation:

2^2x – 5.2^x + 4 = 0

Solve the following equation:

`3^(2x + 3) - 28.3^x + 1 = 0`

Solve the following equation:

`4^(x^2) : 4^x = 16 : 1`

Solve the following equation:

`3^(2x + 1) = 3^(2x - 1) + 216`

If `a^x * b^(3y) = root(4)(a^3 * b^-6)`, find the values of x and y, where a and b are different positive prime numbers.

Solve:

`sqrt((4/3)^(1 - 3x)) = 2 10/27`

Solve:

`sqrt((2/5)^(4x - 3)) = 15 5/8`

If `2^(3x) = (root(3)(32))^(4/y) = (sqrt(8))^5`, then find the values of x and y.

If `5^(2x - 1) = 5^(2x - 2) + 100`, find the value of (2x)^3x.

If `3^(3x + 1) = 3^(3x - 1) + 72`, find the value of `(3x + 1)^(2x)`.

Solve for x and y:

`8^(3 - x/2) - 2^(2y) = 0` and `(sqrt(32))^x ÷ 2^y = 4`

x⁰ is equal to ______.

a⁵ × a^–5 is equal to ______.

`(x^4 xx x^3)/x^9` is equal to ______.

If x = 3, y = 2, then the value of x^x + y^y is ______.

(81)^0.18 × (81)^0.07 is equal to ______.

If `2^(2x + 2) - 5 * 2^x + 1 = 0`, then the value of x is ______.

If 2^x · 3^y = 288, then the value of y is ______.

If a = b^x, b = c^y, c = a^z, then the value of 8 xyz is ______.

`(a^x/(a^-y))^(x - y) (a^y/a^-z)^(y - z) (a^z/a^-x)^(z - x)` is equal to ______.

`(3^(n + 1) xx 9^(n - 1))/(3^n xx 9^(n + 1))` is equal to ______.

(i) lf x = 2, y = 1 then x^x + y^y = 5.

(ii) If a = b^x, b = c^y, c = a^z then xyz = 1.

(i) lf m ≠ n and (m + n)^–1 (m^–1 + n^–1) = m^xn^y then x + y = 2.

(ii) If 2^x = 3^y = 108^z then `2/y + 3/z = 1/x`

(i) `(a^x/a^y)^(x + y) * (a^y/a^z)^(y + z) * (a^z/a^x)^(z + x) = 1`

(ii) a^m × aⁿ = a^{m – n}

(i) `(64)^(-3//2) = 512`

(ii) `5^-1 xx 3^0 = 1/5`