Frank solutions for Mathematics [English] Class 9 ICSE chapter 9 - Indices [Latest edition]

Evaluate the following : 6°

Evaluate the following: `(1/2)^-3`

Evaluate the following: `(2^3)^2`

Evaluate the following: `(3^2)^2`

Evaluate the following: `(0.008)^(2/3)`

Evaluate the following: `(0.00243)^(-3/5)`

Evaluate the following: `root(6)(25^3)`

Evaluate the following: `(2 10/27)^(2/3)`

Evaluate the following:

`9^4 ÷ 27^(-2/3)`

Evaluate the following:

`7^-4 xx (343)^(2/3) ÷ (49)^(-1/2)`

Evaluate the following:

`(64/216)^(2/3) xx (16/36)^(-3/2)`

Write each of the following in the simplest form:
(a³)⁵ x a⁴

Write each of the following in the simplest form:
a² x a³ ÷ a⁴

Write each of the following in the simplest form:

`"a"^(1/3) ÷ "a"^(-2/3)`

Write each of the following in the simplest form:
a^-3 x a² x a⁰

Write the following in the simplest form:
(b^-2 - a^-2) ÷ (b^-1 - a^-1)

Evaluate the following:

`(2^3 xx 3^5 xx 24^2)/(12^2 xx 18^3 xx 27)`

Evaluate the following:

`(4^3 xx 3^7 xx 5^6)/(5^8 xx 2^7 xx 3^3)`

Evaluate the following:

`(12^2 xx 75^-2 xx 35 xx 400)/(48^2 xx 15^-3 xx 525)`

Evaluate the following:

`(2^6 xx 5^-4 xx 3^-3 xx 4^2)/(8^3 xx 15^-3 xx 25^-1)`

Simplify the following and express with positive index:
3p^-2q³ ÷ 2p³q^-2

Simplify the following and express with positive index:

`[("p"^-3)^(2/3)]^(1/2)`

Evaluate the following:

`(1 - 15/64)^(-1/2)`

Evaluate the following:

`(8/27)^((-2)/3) - (1/3)^-2 - 7^0`

Evaluate the following:

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

Evaluate the following:

`(27)^(2/3) xx 8^((-1)/6) ÷ 18^((-1)/2)`

Evaluate the following:

`16^(3/4) + 2(1/2)^-1 xx 3^0`

Evaluate the following:

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

Simplify the following:

`(27 xx^9)^(2/3)`

Simplify the following:

`(8 xx^6y^3)^(2/3)`

Simplify the following:

`((64"a"^12)/(27"b"^6))^(-2/3)`

Simplify the following:

`((36"m"^-4)/(49"n"^-2))^(-3/2)`

Simplify the following:

`("a"^(1/3) + "a"^(-1/3))("a"^(2/3) - 1 + "a"^(-2/3))`

Simplify the following:

`root(3)(x^4y^2) ÷ root(6)(x^5y^-5)`

Simplify the following:

`{("a"^"m")^("m" - 1/"m")}^(1/("m" + 1)`

Simplify the following:
`x^("m" + 2"n"). x^(3"m" - 8"n") ÷ x^(5"m" - 60)`

Simplify the following:

`(81)^(3/4) - (1/32)^(-2/5) + 8^(1/3).(1/2)^-1. 2^0`

Simplify the following:

`(27/343)^(2/3) ÷ (1)/(625/1296)^(1/4) xx (536)/root(3)(27)`

Simplify the following:

`(5^x xx 7 - 5^x)/(5^(x + 2) - 5^(x + 1)`

Simplify the following:

`(3^(x + 1) + 3^x)/(3^(x + 3) - 3^(x + 1)`

Simplify the following:

`(2^"m" xx 3 - 2^"m")/(2^("m" + 4) - 2^("m" + 1)`

Simplify the following:

`(5^("n" + 2) - 6.5^("n" + 1))/(13.5^"n" - 2.5^("n" + 1)`

Solve for x:
2^2x+1= 8

Solve for x:
3 x 7^x = 7 x 3^x

Solve for x:
2^{x + 3} + 2^{x + 1} = 320

Solve for x:
`9 xx 3^x = (27)^(2x - 5)`

Solve for x:
2^2x+3 - 9 x 2^x + 1 = 0

Solve for x:
1 = p^x

Solve for x:
p³ x p^-2 = p^x

Solve for x:

`"p"^-5 = (1)/"p"^(x + 1)`

Solve for x:
2^2x + 2^{x +2} - 4 x 2³ = 0

Solve for x:

9 x 81^x = `(1)/(27^(x - 3)`

Solve for x:
2^{2x- 1} − 9 x 2^{x − 2} + 1 = 0

Solve for x:
5^x2 : 5^x = 25 : 1

Solve for x:

`sqrt((8^0 + 2/3)` = (0.6)^2-3x 

Solve for x:

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

Solve for x:
9^x+4 = 3² x (27)^x+1

Find the value of k in each of the following:

`(root(3)(8))^((-1)/(2)` = 2^k 

Find the value of k in each of the following:

`root(4)root(3)(x^2)` = x^k 

Find the value of k in each of the following:

`(sqrt(9))^-7 xx (sqrt(3))^-5` = 3^k 

Find the value of k in each of the following:

`(1/3)^-4 ÷ 9^((-1)/(3)` = 3^k 

If a = `2^(1/3) - 2^((-1)/3)`, prove that 2a³ + 6a = 3

If x = `3^(2/3) + 3^(1/3)`, prove that x³ - 9x - 12 = 0

If `root(x)("a") = root(y)("b") = root(z)("c")` and abc = 1, prove that x + y + z = 0

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

Show that : `(1)/(1 + "a"^("p"- "q")) + (1)/(1 + "a"^("q"- "p")`

Find the value of (8p)^p if 9^{p + 2} - 9^p = 240.

If a^x = b^y = c^z and abc = 1, show that

`(1)/x + (1)/y + (1)/z` = 0.

If `x^(1/3) + y^(1/3) + z^(1/3) = 0`, prove that (x + y + z)³ = 27xyz

If 2250 = 2^a. 3^b. 5^c, find a, b and c. Hence, calculate the value of 3^a x 2^-b x 5^-c.

If 2400 = 2^x x 3^y x 5^z, find the numerical value of x, y, z. Find the value of 2^-x x 3^y x 5^z as fraction.

If 2^x = 3^y = 12^z ; show that `(1)/z = (1)/y + (2)/x`.

Find the value of 'a' and 'b' if:

9^2a = `(root(3)(81))^(-6/"b") = (sqrt(27))^2`

Find the value of 'a' and 'b' if:

`(sqrt243)^"a" ÷ 3^("b" + 1)` = 1 and `27^"b" - 81^(4 -"a"/2)` = 0

Prove the following:

`sqrt(x^-1 y) · sqrt(y^-1 z) · sqrt(z^-1 x)` = 1

Prove the following:

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

Prove the following:

`("a"^"m"/"a"^"n")^("m"+"n"+1) ·("a"^"n"/"a"^1)^("n" + 1-"m").("a"^1/"a"^"m")^(1+"m"-"n")`

Prove the following:

`root("ab")(x^"a"/x^"b")·root("bc")(x^"b"/x^"c")·root("ca")(x^"c"/x^"a")` = 1

Prove the following:
(x^a)^b-c x (x^b)^c-a x (x^c)^a-b = 1

Prove the following:

`(x^("p"("q"-"r")))/(x^("q"("p"-"r"))) ÷ (x^"q"/x^"p")^"r"` = 1