Surds And Indices

If x = 3 + 22, then the value of		x	-	1		is:
				x

A:

1

B:

2

C:

22

D:

33

Answer: B

	x	-	1		2	= x +	1	- 2
			x				x

= (3 + 22) +	1	- 2
	(3 + 22)

= (3 + 22) +	1	x	(3 - 22)	- 2
	(3 + 22)		(3 - 22)

= (3 + 22) + (3 - 22) - 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: B

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: D

We know that 11² = 121.

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

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

📖

1	+	1	= ?
1 + a^{(n - m)}		1 + a^{(m - n)}

A:

0

B:

1

2

C:

1

D:

a^{m + n}

Answer: C

1

+

1

=

1

+

1

	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.

📖

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

A:

1

B:

2

C:

9

D:

3ⁿ

Answer: C

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.

📖

Questions for: Surds And Indices