Question

Which of the following expressions would be true for an integer ‘x’ greater than 1, whose square root and cube root are integers a and b respectively?
A. x
2
= x
3

B. (a
2
)
3
= (b
3
)
3

C. x is a perfect square.
D. a is a perfect square.

Answer 1

The expression (a^2)^3 = (b^3)^3 is true

Step-by-step explanation:

Given: Integer ‘x’ greater than 1, whose square root and cube root are integers a and b, respectively.

Solution:

X is positive as it is greater than 1.

Square root of x is a.

Cube root of x is b.

The expressions are as follows:

A. x^2 = x^3 - Cannot be true as a and b are not 1.

B. (a^2)^3 = (b^3)^3

  a^2 = x and b^3 = x

  So (a^2)^3 = (b^3)^3 = x. Hence this expression is true.

C. x is a perfect square.

   - Not necessarily. We just know that a is the square root of a.

D. a is a perfect square.

   - Not necessarily. We just know that a is the square root of a.

So Option B is the answer.

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

An integer ‘x’ greater than 1, whose square root and cube root are integers a and b respectively

TO CHOOSE THE CORRECT OPTIONS

$\sf{ A. \: \: \: {x}^{2} = {x}^{3} \: }$

$\sf{B. \: \: {( {a}^{2} )}^{3} = { ({b}^{3}) }^{3} \: \: }$

$\sf{ C. \: \: x \: is \: a \: perfect \: square\: \: }$

$\sf{D. \: \: a \: \: is \: a \: perfect \: square}$

EVALUATION

By the given condition

$\sf{ \: \sqrt{x} = a\: }$

$\implies \sf{ \:x = {a}^{2} \: } \: \: \: \: ...(1)$

Again

$\sf{ \: \sqrt[3]{x} = b\: }$

$\implies \sf{ \: x = {b}^{3} \: } \: \: \: ........(2)$

CHECKING FOR OPTION : A

This is possible only when x = 1

But by the given condition x > 1

So OPTION A is not possible

CHECKING FOR OPTION : B

$\sf{ \: \: {( {a}^{2} )}^{3} = { ({b}^{3}) }^{3} \: \: }$

Using Equation (1) & Equation (2) we get

$\sf{ \: {x}^{3} = {x}^{3} \: }$

Which is true for all values of x

So OPTION B is CORRECT

CHECKING FOR OPTION : C

$\sf{ Since \: \: \:x = {a}^{2} \: } \: \: \: \:$

So x is a perfect square

So OPTION C is CORRECT

CHECKING FOR OPTION : D

For example x = 64

$\sf{ \: Then \: \: a = 8 \: \: and \: \: b= 4 \: }$

So a is not necessarily a perfect square

So OPTION D is NOT CORRECT

RESULT

OPTION B & OPTION C are CORRECT

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