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The expression \( \Large \left[ \left(\sqrt{2}\right)^{\sqrt{2}} \right]^{\sqrt{2}} \)

A) a natural number
B) a integer and not a natural number
C) a rational number but not an integer
D) a real number but not a rational number

Correct Answer:




D) a real number but not a rational number

Description for Correct answer:

Given expression = \( \Large \left[ \left(\sqrt{2}\right)^{\sqrt{2}} \right]^{\sqrt{2}} \)

= \( \Large \left(\sqrt{2}\right)^{ \left(2\right)^{\frac{\sqrt{2}}{2}} } = \left(\sqrt{2}\right)^{ \left(2\right)^{\frac{1}{\sqrt{2}}} } \)

= \( \Large \left(2\right)^{\frac{1}{2} \times 2^{\frac{1}{\sqrt{2}}}} = 2^{ \left(\frac{2}{2}^{\frac{1}{\sqrt{2}}}\right) } = \left(2\right)^{ \left(2\right)^{ \left(\frac{1}{\sqrt{2}}-1\right) } } \)

which denotes a real number but not a rational number.

Part of solved Indices and Surd questions and answers : >> Elementary Mathematics >> Indices and Surd

Comments

Similar Questions

1). If \( \Large 16 \times 8^{n+2} = 2^{m} \), then m is equal to

A). n+8

B). 2n+10

C). 3n+2

D). 3n+10

2). If \( \Large \sqrt{10 + \sqrt[3]{x}} = 4 \), then what is the value of x ?

A). 150

B). 216

C). 316

D). 450

3). If m and n are natural numbers, then \( \Large \sqrt[m]{n} \) is

A). always irrational

B). irrational unless n is the mth power of an integer

C). irrational unless m is the nth power of an integer

D). irrational unless m and n are coprime

4). Consider the following in respect of the numbers \( \Large \sqrt{2}, \sqrt[3]{3} \) and \( \Large \sqrt[6]{6} \)
I. \( \Large \sqrt[6]{6} \) is the greatest number.
II. \( \Large \sqrt{2} \) is the smallest number.
Which of the above statements is/are correct?

A). Only I

B). Only II

C). Both I and II

D). Neither I nor II

5). If \( \Large a = \frac{\sqrt{3}}{2} \),then \( \Large \sqrt{1 + a} + \sqrt{1 - a} \) = ?

A). \( \Large \left(2 - \sqrt{3}\right) \)

B). \( \Large \left(2 + \sqrt{3}\right) \)

C). \( \Large \frac{\sqrt{3}}2{} \)

D). \( \Large \sqrt{3} \)

6). Simplify \( \Large \sqrt[6]{ \left(27\right)^{-\frac{2}{3}} } + \left(8\right)^{-\frac{2}{3}} \)

A). \( \Large \sqrt[6]{35} \)

B). \( \Large \frac{6}{\sqrt{13}} \)

C). \( \Large \sqrt{13} \)

D). \( \Large \sqrt[6]{6} \)

7). If \( \Large 2x^{\frac{1}{3}} + 2x^{-\frac{1}{3}} = 5 \), then \( \Large x^{\frac{1}{3}} \) is equal to

A). 1 or -1

B). 2 or \( \Large \frac{1}{2} \)

C). 8 or \( \Large \frac{1}{8} \)

D). 3 or \( \Large \frac{1}{3} \)

8). If \( \Large log3 \times log4\ x > 0 \), then

A). \( \Large x > 1 \)

B). \( \Large x > 4 \)

C). \( \Large x > 64 \)

D). none of these

9). \( \Large log \frac{1}{4} \left(a^{2}-1\right) < log_\frac{1}{2} \left(a+1\right)^{2}\)

A). \( \Large a<1 \)

B). \( \Large a<-1 \)

C). \( \Large a>1 \)

D). none of these

10). If \( \Large log_{e} \left(\frac{a+b}{2}\right) = \frac{1}{2} \left(log_{e} a+log_{e} b\right) \) then:

A). a = b

B). a = b/2

C). 2a = b

D). \( \Large a = \frac{b}{3} \)