Question

Consider the expression below: 1/√(k+7) Given that the expression represents a rational number, which of these is the least possible positive integer value of k
a) 0
b)2 c) 8
d)10

Answer 1

Answer:

b)2

Step-by-step explanation:

Since 1/√(k+7) is a rational number.

√(k+7) must be rational.

From the given options, option b satisfies the condition for the expression to be a rational number.

Answer 2

The least possible positive integer value of k = 2 [ The correct option is b) 2 ]

Given :

• Consider the expression : 1/√(k+7)

• Given that the expression represents a rational number

To find :

The least possible positive integer value of k is

a) 0

b) 2

c) 8

d) 10

Concept :

Rational number :

A rational number is defined as a number of the form p/q where p & q are integers with q ≠ 0

Examples :

$\displaystyle \sf{2,- 1, \frac{1}{3} , - \frac{12}{23}} \: are \: the \: examples \: of \: rational \: numbers$

Solution :

Step 1 of 2 :

Write down the given expression

Here the given expression is

$\displaystyle \sf = \frac{1 }{ \sqrt{k + 7} }$

Step 2 of 2 :

Find the least possible positive integer value of k

The expression

$\displaystyle \sf = \frac{1 }{ \sqrt{k + 7} }$

Given that the expression represents a rational number

We know that a rational number is defined as a number of the form p/q where p & q are integers with q ≠ 0

$\displaystyle \sf \therefore \: \sqrt{k + 7} \: \: must \: be \: an \: integer$

$\displaystyle \sf{ \implies }{(k + 7)} \: \: must \: be \: a \: perfect \: square$

Check for option (a)

For k = 0 we have (k + 7) = 7 which is not a perfect square.

So option (a) is not correct.

Check for option (b)

For k = 2 we have (k + 7) = 9 which is a perfect square.

So option (b) is correct.

Check for option (c)

For k = 8 we have (k + 7) = 15 which is not a perfect square.

So option (c) is not correct.

Check for option (d)

For k = 10 we have (k + 7) = 17 which is not a perfect square.

So option (d) is not correct.

Hence the correct option is b) 2

∴ The least possible positive integer value of k = 2

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