#### The square root of a positive integer is an irrational number.

Consider √9=3

Hence, the square root of a positive integer need not be an irrational number.

#### The value of ^{5 }**√**16×^{5}**√**2 is ____.

**√**

**√**

^{5}* √*16×

^{5}

*2=*

**√**^{5}

**√***16×2)*

**(**=

^{5}

*32*

**√**=

^{5}

*2*

**√**^{5}

= 2

Hence, the value of

^{5}

*16×*

**√**^{5}

*2 is 2.*

**√**#### (5+2√3)(5−3√3)=____.

(5+2√3)(5−3√3)=5×5−5×3√3+√3×5−2√3×3√3

=25−15√3+10√3−18

=7−5√3

∴(5+2√3)(5−3√3)=7−5√3

#### 2^{8 }÷^{ }2^{3 }= ________.

2^{8 }÷^{ }2^{3 }= 2^{8 }/ 2^{3}

= 2^{8-3 }( a^{m }/a^{n }=a^{m-n })

= 2^{5}

=32

2^{8}÷^{}2^{3 }=32

#### Which of the following is a rational number?

0.3333333... is a non-terminating and recurring decimal.

We know that,

a non-terminating and recurring decimal can be converted to the form of p/q, where p, q are integers and q ≠ 0.

0.3333333...=1/3

Hence, 0.3333333... is a rational number.

#### The number (2− √2)(2+ √2) is an irrational number.

(2−√2)(2+√2)=22−(* √*2)

^{2 }

= 4 − 2

= 2

Hence, the number (2−√2)(2+√2) is a rational number.

#### (2√3+4√2)+( √3−2√2)=____.

(2√3+4√2)+( √3−2√2)=2√3+√3+4√2−2√2

=3√3+2√2

∴(2√3+4√2)+( √3−2√2)=3√3+2√2

#### The number of rational numbers between two natural numbers is infinite.

The number of rational numbers between two natural numbers is infinite.

#### √ (2025) is an irrational number.

√2025=√(25×81)

= 5 × 9

= 45, which is a rational number.

Hence, √2025 is a rational number.

#### If 2^{n-1 }+ 2 ^{n+1 }= 320, then n =_______

Given, 2^{n-1 }+ 2^{n+1 }= 320

2^{n-1 }(1 + 2^{2 })= 320

5 x 2^{n-1 }=320

2^{n-1} = 320/5

2^{n-1 }= 64

2^{n-1 }= 2^{6}

n-1 = 6

n= 7
Given, 2^{n-1 }+ 2^{n+1 }= 320

2^{n-1 }(1 + 2^{2 })= 320

5 x 2^{n-1 }=320

2^{n-1} = 320/5

2^{n-1 }= 64

2^{n-1 }= 2^{6}

n-1 = 6

n= 7

#### The product of 3 and √5 is a rational number.

The product of a rational number and an irrational number is irrational as long as the rational is not zero.

3×√5=3√5,which is an irrational.

Hence, the product of 3 and √5 is an irrational number.

#### The solutions of the equation x^{2}−9=0 are irrational numbers.

Given equation is x^{2}−9=0.

⇒ x^{2}=9

⇒x=√9

⇒x=±3

3 and −3 are rational numbers.

Hence, the solutions of x^{2}−9=0 are rational numbers.

#### If √5 = 2.236, then the value of 2/√5 up to three places of decimals is____.

2/√5 = 2/√5 X √5/√5

= 2/√5/5

= 2 X 2.236/5 (Given √5 = 2.236)

= 4.472/5

= 0.8944
Hence, the required three decimals value is 0.8944.

#### The number 3−√3 is a rational number.

The difference of a rational number and an irrational number is an irrational number.

Hence, the number 3−√3 is an irrational number.

#### 3^{2 }x 3^{3} =_______.

3^{2 }x 3^{3} = 3^{2+3 }( a^{m }x a^{n }=a^{m+n})

= 3^{5}

= 243

3^{2 }x 3^{3}= 243

#### Every whole number is a natural number.

'0' is a whole number but not a natural number.

Hence, every whole number except 0 is a natural number.

#### The sum of a rational number and an irrational number is an irrational number.

Let us check this with an example:

Consider an irrational number √2= 1.41421...

The sum of 2 and √2 is again a non-terminating and non-recurring decimal number.

Hence, the sum of a rational number and an irrational number is an irrational number.

#### A non-terminating and non-recurring decimal is known as an irrational number.

A non-terminating and non-recurring decimal is known as an irrational number.

#### The product of two irrational numbers is always irrational.

Consider, √2×√8=√16

= 4

Hence, the product of two irrational numbers need not be an irrational number.

#### A number of the form p/q is called a rational number.

A number of the form p/q is called a rational number.

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