Surds and Indices - General Questions

If x = 3 + 2√2, then the value of		√x	-	1		is:
				√x

A.	1
B.	2
C.	2√2
D.	3√3

Answer: Option B

Explanation:

	√x	-	1		2	= x +	1	- 2
			√x				x

= (3 + 22) +	1	- 2
	(3 + 2√2)

= (3 + 2√2) +	1	x	(3 - 2√2)	- 2
	(3 + 2√2)		(3 - 2√2)

= (3 + 2√2) + (3 - 2√2) - 2

= 4.

		√x	-	1		= 2.
				√x

x^b

(b + c - a)

x^c

(c + a - b)

x^a

^{(a + b - c)}

= ?

x^c

x^a

x^b

A.	x abc
B.	1
C.	x ab + bc + ca
D.	x a + b + c

Answer: Option B

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

If m and n are whole numbers such that mn = 121, the value of (m - 1)n + 1 is:

A.	1
B.	10
C.	121
D.	1000

Answer: Option D

Explanation:

We know that 11² = 121.

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

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

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

9ⁿ x 3^{n - 1}

A.	1
B.	2
C.	9
D.	3n

Answer: Option C

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.

(0.04)^-1.5 = ?

A.	25
B.	125
C.	250
D.	625

Answer: Option B

Explanation:

(0.04)^-1.5 =		4		-1.5
		100

=		1		-(3/2)
		25

= (25)^(3/2)

= (5²)^(3/2)

= (5)^{2 x (3/2)}

= 5³

= 125.

(25)^7.5 x (5)^2.5 ÷ (125)^1.5 = 5^?

A.	8.5
B.	13
C.	16
D.	17.5
E.	None of these

Answer: Option B

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.

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

A.	0
B.	1
C.	xa - b - c
D.	None of these

Answer: Option B

Explanation:

Given Exp. =

	1 +	x^b	+	x^c
		x^a		x^a

	1 +	x^a	+	x^c
		x^b		x^b

	1 +	x^b	+	x^a
		x^c		x^c

=	x^a	+	x^b	+	x^c
	(x^a + x^b + x^c)		(x^a + x^b + x^c)		(x^a + x^b + x^c)

=	(x^a + x^b + x^c)
	(x^a + x^b + x^c)

The value of [(10)¹⁵⁰ ÷ (10)¹⁴⁶]

A.	1000
B.	10000
C.	100000
D.	10 6

Answer: Option B

Explanation:

(10)¹⁵⁰ ÷ (10)¹⁴⁶ =	10¹⁵⁰
	10¹⁴⁶

= 10^{150 - 146}

= 10⁴

= 10000.

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