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Surds and Indices problems

1. If 3^{(x - y)} = 27 and 3^{(x + y)} = 243, then x is equal to:
Options
A. 0
B. 2
C. 4
D. 6

Discuss

Correct Answer: 4

Explanation:

3 ^{x - y} = 27 = 3 ³ ⟺ x - y = 3 ....(i)

3^{x + y} = 243 = 3⁵ ⟺ x + y = 5 ....(ii)

On solving (i) and (ii), we get x = 4.

2. (25)^7.5 x (5)^2.5 � (125)^1.5 = 5^?
Options
A. 8.5
B. 13
C. 16
D. 17.5
E. None of these

Discuss

Correct Answer: 13

Explanation:

Let (25) ^7.5 x (5) ^2.5 � (125) ^1.5 = 5 ^x

Then,	(5²)^7.5 x (5)^2.5	= 5^x
	(5³)^1.5

⟹	5^{(2 x 7.5)} x 5^2.5	= 5x
	5^{(3 x 1.5)}

⟹	5¹⁵ x 5^2.5	= 5^x
	5^4.5

⟹ 5^x = 5^{(15 + 2.5 - 4.5)}

⟹ 5^x = 5¹³

∴ x = 13.

❨

x^b

❩

(b + c - a)

❨

x^c

❩

(c + a - b)

❨

x^a

❩

^{(a + b - c)}

x^c

x^a

x^b

Options
A. x^abc
B. 1
C. x^{ab + bc + ca}
D. x^{a + b + c}

Discuss

Correct Answer: 1

Explanation:

Given Exp.

= x^{(b - c)(b + c - a)} . x^{(c - a)(c + a - b)} . x^{(a - b)(a + b - c)}

= x^{(b - c)(b + c) - a(b - c)} . x^{(c - a)(c + a) - b(c - a)}
. x^{(a - b)(a + b) - c(a - b)}

= x^{(b² - c² + c² - a² + a² - b²)} . x^{-a(b - c) - b(c - a) - c(a - b)}

= (x⁰ x x⁰)

= (1 x 1) = 1.

4. If m and n are whole numbers such that mⁿ = 121, the value of (m - 1)^{n + 1} is:
Options
A. 1
B. 10
C. 121
D. 1000

Discuss

Correct Answer: 1000

Explanation:

We know that 11 ² = 121.

Putting m = 11 and n = 2, we get:

(m - 1)^{n + 1} = (11 - 1)^{(2 + 1)} = 10³ = 1000.

5. Given that 10^0.48 = x, 10^0.70 = y and x^z = y², then the value of z is close to:
Options
A. 1.45
B. 1.88
C. 2.9
D. 3.7

Discuss

Correct Answer: 2.9

Explanation:

x ^z = y ² ⟺ 10 ^(0.48z) = 10 ^{(2 x 0.70)} = 10 ^1.40

⟹ 0.48z = 1.40

⟹ z =	140	=	35	= 2.9 (approx.)
	48		12

6.

If ❨ a ❩ x - 1 = ❨ b ❩ x - 3 , then the value of x is:

b a
Options
A. ¹⁄₂
B. 1
C. 2
D. ⁷⁄₂

Discuss

Correct Answer: 2

Explanation:

Given ❨	a	❩	x - 1	=	❨	b	❩	x - 3
	b					a

⟹

❨

❩

x - 1

❨

❩

-(x - 3)

❨

❩

(3 - x)

⟹ x - 1 = 3 - x

⟹ 2x = 4

⟹ x = 2.

7.

(243)^n/5 x 3^{2n + 1} =?

9ⁿ x 3^{n - 1}
Options
A. 1
B. 2
C. 9
D. 3ⁿ

Discuss

Correct Answer: 9

Explanation:

Given Expression

=	(243)^(n/5) x 3^{2n + 1}
	9ⁿ x 3^{n - 1}

=	(3⁵)^(n/5) x 3^{2n + 1}
	(3²)ⁿ x 3^{n - 1}

=	(3^{5 x (n/5)} x 3^{2n + 1})
	(3²ⁿ x 3^{n - 1})

=	3ⁿ x 3^{2n + 1}
	3²ⁿ x 3^{n - 1}

=	3^{(n + 2n + 1)}
	3^{(2n + n - 1})

=	3^{3n + 1}
	3^{3n - 1}

= 3^{(3n + 1 - 3n + 1)} = 3² = 9.

8.

1 + 1 =?

1 + a^{(n - m)} 1 + a^{(m - n)}
Options
A. 0
B. 1¹⁄₂
C. 1
D. a^{m + n}

Discuss

Correct Answer: 1

Explanation:

❨	1 +	aⁿ	❩
		a^m

❨	1 +	a^m	❩
		aⁿ

1 + a^{(n - m)}

1 + a^{(m - n)}

=	a^m	+	aⁿ
	(a^m + aⁿ)		(a^m + aⁿ)

=	(a^m + aⁿ)
	(a^m + aⁿ)

= 1.

9. The value of [(10)¹⁵⁰ � (10)¹⁴⁶]
Options
A. 1000
B. 10000
C. 100000
D. 10⁶

Discuss

Correct Answer: 10000

Explanation:

(10)¹⁵⁰ � (10)¹⁴⁶ =	10¹⁵⁰
	10¹⁴⁶

= 10^{150 - 146}

= 10⁴

= 10000.

10. If 5^a = 3125, then the value of 5^{(a - 3)} is:
Options
A. 25
B. 125
C. 625
D. 1625

Discuss

Correct Answer: 25

Explanation:

5 ^a = 3125 ⟺ 5 ^a = 5 ⁵

⟹ a = 5.

∴ 5^{(a - 3)} = 5^{(5 - 3)} = 5² = 25.

First 2 3 4 5 6 7 8

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