# Frank solutions for Class 9 Maths ICSE chapter 9 - Indices [Latest edition]

## Chapter 9: Indices

Exercise 9.1
Exercise 9.1

### Frank solutions for Class 9 Maths ICSE Chapter 9 Indices Exercise 9.1

Exercise 9.1 | Q 1.01

Evaluate the following : 6°

Exercise 9.1 | Q 1.02

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

Exercise 9.1 | Q 1.03

Evaluate the following: (2^3)^2

Exercise 9.1 | Q 1.04

Evaluate the following: (3^2)^2

Exercise 9.1 | Q 1.05

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

Exercise 9.1 | Q 1.06

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

Exercise 9.1 | Q 1.07

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

Exercise 9.1 | Q 1.08

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

Exercise 9.1 | Q 2.1

Evaluate the following:

9^4 ÷ 27^(-2/3)

Exercise 9.1 | Q 2.2

Evaluate the following:

7^-4 xx (343)^(2/3) ÷ (49)^(-1/2)

Exercise 9.1 | Q 2.3

Evaluate the following:

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

Exercise 9.1 | Q 3.1

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

Exercise 9.1 | Q 3.2

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

Exercise 9.1 | Q 3.3

Write each of the following in the simplest form:

"a"^(1/3) ÷ "a"^(-2/3)

Exercise 9.1 | Q 3.4

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

Exercise 9.1 | Q 3.5

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

Exercise 9.1 | Q 4.1

Evaluate the following:

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

Exercise 9.1 | Q 4.2

Evaluate the following:

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

Exercise 9.1 | Q 4.3

Evaluate the following:

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

Exercise 9.1 | Q 4.4

Evaluate the following:

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

Exercise 9.1 | Q 5.1

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

Exercise 9.1 | Q 5.2

Simplify the following and express with positive index:

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

Exercise 9.1 | Q 6.1

Evaluate the following:

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

Exercise 9.1 | Q 6.2

Evaluate the following:

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

Exercise 9.1 | Q 6.3

Evaluate the following:

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

Exercise 9.1 | Q 6.4

Evaluate the following:

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

Exercise 9.1 | Q 6.5

Evaluate the following:

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

Exercise 9.1 | Q 6.6

Evaluate the following:

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

Exercise 9.1 | Q 7.01

Simplify the following:

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

Exercise 9.1 | Q 7.02

Simplify the following:

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

Exercise 9.1 | Q 7.03

Simplify the following:

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

Exercise 9.1 | Q 7.04

Simplify the following:

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

Exercise 9.1 | Q 7.05

Simplify the following:

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

Exercise 9.1 | Q 7.06

Simplify the following:

root(3)(x^4y^2) ÷ root(6)(x^5y^-5)

Exercise 9.1 | Q 7.07

Simplify the following:

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

Exercise 9.1 | Q 7.08

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

Exercise 9.1 | Q 7.09

Simplify the following:

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

Exercise 9.1 | Q 7.1

Simplify the following:

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

Exercise 9.1 | Q 8.1

Simplify the following:

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

Exercise 9.1 | Q 8.2

Simplify the following:

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

Exercise 9.1 | Q 8.3

Simplify the following:

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

Exercise 9.1 | Q 8.4

Simplify the following:

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

Exercise 9.1 | Q 9.01

Solve for x:
22x+1= 8

Exercise 9.1 | Q 9.02

Solve for x:
3 x 7x = 7 x 3x

Exercise 9.1 | Q 9.03

Solve for x:
2x + 3 + 2x + 1 = 320

Exercise 9.1 | Q 9.04

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

Exercise 9.1 | Q 9.05

Solve for x:
22x+3 - 9 x 2x + 1 = 0

Exercise 9.1 | Q 9.06

Solve for x:
1 = px

Exercise 9.1 | Q 9.07

Solve for x:
p3 x p-2 = px

Exercise 9.1 | Q 9.08

Solve for x:

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

Exercise 9.1 | Q 9.09

Solve for x:
22x + 2x +2 - 4 x 23 = 0

Exercise 9.1 | Q 9.1

Solve for x:

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

Exercise 9.1 | Q 9.11

Solve for x:
22x- 1 -9 x 2x - 2 + 1= 0

Exercise 9.1 | Q 9.12

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

Exercise 9.1 | Q 9.13

Solve for x:

sqrt((8^0 + 2/3) = (0.6)2-3x

Exercise 9.1 | Q 9.14

Solve for x:

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

Exercise 9.1 | Q 9.15

Solve for x:
9x+4 = 32 x (27)x+1

Exercise 9.1 | Q 10.1

Find the value of k in each of the following:

(root(3)(8))^((-1)/(2) = 2k

Exercise 9.1 | Q 10.2

Find the value of k in each of the following:

root(4)root(3)(x^2) = xk

Exercise 9.1 | Q 10.3

Find the value of k in each of the following:

(sqrt(9))^-7 xx (sqrt(3))^-5 = 3k

Exercise 9.1 | Q 10.4

Find the value of k in each of the following:

(1/3)^-4 ÷ 9^((-1)/(3) = 3k

Exercise 9.1 | Q 11

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

Exercise 9.1 | Q 12

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

Exercise 9.1 | Q 13

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

Exercise 9.1 | Q 14

If ax = by = cz and b2 = ac, prove that y = (2xz)/(z + x)

Exercise 9.1 | Q 15

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

Exercise 9.1 | Q 16

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

Exercise 9.1 | Q 17

If ax = by = cz and abc = 1, show that

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

Exercise 9.1 | Q 18

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

Exercise 9.1 | Q 19

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

Exercise 9.1 | Q 20

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

Exercise 9.1 | Q 21

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

Exercise 9.1 | Q 22.1

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

92a = (root(3)(81))^(-6/"b") = (sqrt(27))^2

Exercise 9.1 | Q 22.2

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

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

Exercise 9.1 | Q 23.1

Prove the following:

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

Exercise 9.1 | Q 23.2

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

Exercise 9.1 | Q 23.3

Prove the following:

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

Exercise 9.1 | Q 23.4

Prove the following:

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

Exercise 9.1 | Q 23.5

Prove the following:
(xa)b-c x (xb)c-a x (xc)a-b = 1

Exercise 9.1 | Q 23.6

Prove the following:

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

Exercise 9.1

## Frank solutions for Class 9 Maths ICSE chapter 9 - Indices

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Concepts covered in Class 9 Maths ICSE chapter 9 Indices are Handling Positive, Fraction, Negative and Zero Indices, Simplification of Expressions, Solving Exponential Equations, Laws of Exponents.

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