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Right option is (d) 6

Explanation: We know that ACCORDING to LAWS of EXPONENTS, 1/18^-1/2 = 18^1/2

Hence, (2)^1/2/(18)^-1/2 =(2)^1/2 (18)^1/2

According to laws of exponents, (a^m)*(b^m) = (ab)^m

Applying that rule here, (2)^1/2 (18)^1/2 = (2*18)^1/2

= (36)^1/2

= 6.

Correct ANSWER is (c) 1000

To EXPLAIN: According to laws of exponents, (a^m)*(b^m) = (AB)^m

Applying that rule here, 2^3*5^3=(2*5)^3

=(10)^3

= 1000.

Right OPTION is (a) 4

The best I can explain: ACCORDING to LAWS of exponents, (a^m)*(b^m) = (ab)^m

Applying that rule here, (4)^1/3*(16)^1/3 = (4*16)^1/3

= (64)^1/3

= 4.

The CORRECT option is (b) 13^-3

For explanation: ACCORDING to laws of exponents, a^m/a^n = a^m-n

Applying that rule here, (13)^1/2/(13)^7/2 = 13^(1/2-7/2)

= 13^((1)-(7))/(2)

= 13^((-6))/(2)

= 13^-3.

Correct answer is (B) 4^6

Best EXPLANATION: According to laws of exponents, a^m/a^n = a^m-n

Applying that RULE here, (4)^9/(4)^3 = 4^(9-3)

= 4^6.

The CORRECT choice is (d) \(\sqrt[2]{5}\)

To elaborate: ACCORDING to laws of exponents, a^m/a^n = a^m-n

Applying that RULE here, (5)^3/4/(5)^1/4 = (5)^(3/4-1/4)

= (5)^(3-1)/4

= (5)^2/4

= (5)^1/2

= \(\sqrt[2]{5}\).

Right OPTION is (a) 81

To explain: According to LAWS of exponents, a^m a^n = a^(m+n)

APPLYING that rule here, (9)^1/4 * (9)^7/4 = (9)^(1+7)/4

= (9)^8/4

= (9)^2

= 81.

Correct ANSWER is (c) (7)^23/20

For explanation I WOULD say: According to laws of exponents, a^m a^n = a^(m+n)

APPLYING that rule here, (7)^2/5 * (7)^3/4 = (7)^(2/5+3/4)

= (7)^((2*4)+(5*3))/(5*4)

= (7)^(8+15)/20

= (7)^23/20.

The correct option is (c) \(\sqrt{27}\)

The EXPLANATION is: ACCORDING to laws of exponents, a^m/n = \((\sqrt[n]{a})^m\).

APPLYING that rule here, 3^3/2 = (3^3)^1/2

= 27^1/2

= \(\sqrt{27}\).

Correct OPTION is (a) 4

For EXPLANATION: According to laws of exponents, a^m/n = \((\sqrt[n]{a})^m\).

Applying that rule here, 8^2/3 = \((\sqrt[3]{8})^2\)

= 2^2

= 4.

Right choice is (b) \(\frac{5-\sqrt{7}}{18}\)

Explanation: When the denominator of an expression contains a term with a SQUARE root, the process of CONVERTING it to an equivalent expression whose denominator is a rational number is called rationalising the denominator.

By MULTIPLYING \(\frac{1}{5+\sqrt{7}}\) by \(5-\sqrt{7}\), we will GET same expression since \(\frac{5-\sqrt{7}}{5-\sqrt{7}}\) = 1.

Therefore, \(\frac{1}{5+\sqrt{7}} = {\frac{1}{5+\sqrt{7}}} * (\frac{5-\sqrt{7}}{5-\sqrt{7}})\)

= \(\frac{5-\sqrt{7}}{(5*5)-(\sqrt{7}*\sqrt{7})}\)

= \(\frac{5-\sqrt{7}}{25-7}\)

= \(\frac{5-\sqrt{7}}{18}\).

Right answer is (d) \(\frac{6+\sqrt{3}}{33}\)

BEST explanation: When the denominator of an expression contains a term with a square root, the process of converting it to an EQUIVALENT expression whose denominator is a rational number is called rationalising the denominator.

By MULTIPLYING \(\frac{1}{(6-\sqrt{3})}\) by \(6+\sqrt{3}\), we will GET same expression since \(\frac{6+\sqrt{3}}{6+\sqrt{3}}\) = 1.

Therefore, \(\frac{1}{(6-\sqrt{3})} = \frac{1}{(6-\sqrt{3})} * \frac{6+\sqrt{3}}{6+\sqrt{3}} = \frac{6+\sqrt{3}}{(6*6) – (\sqrt{3}*\sqrt{3})}\)

= \(\frac{6+\sqrt{3}}{(36-3)}\)

= \(\frac{6+\sqrt{3}}{33}\).

Correct option is (b) 5 + 2√6

Explanation: We KNOW that \((\sqrt{a}+\sqrt{b}) * (\sqrt{a}+\sqrt{b}) = a + 2\sqrt{a}\sqrt{b} + b.\)

By applying that rule here, we GET \((\sqrt{3}+\sqrt{2}) * (\sqrt{3}+\sqrt{2}) = 3 + 2\sqrt{3}\sqrt{2} + 2

= 5 + 2\sqrt{6}\) SINCE \(\sqrt{a}\sqrt{b} = \sqrt{(ab)}\),

\(\sqrt{3}\sqrt{2} = \sqrt{(3*2)} = \sqrt{6}\).

The correct option is (a) \(\frac{\SQRT{3}}{3}\)

EXPLANATION: When the denominator of an expression contains a term with a square root, the process of converting it to an equivalent expression WHOSE denominator is a rational number is CALLED rationalising the denominator.

By multiplying \(\frac{1}{\sqrt{3}}\) by \(\frac{\sqrt{3}}{\sqrt{3}}\), we will get same expression SINCE \(\frac{\sqrt{3}}{\sqrt{3}}\) = 1.

Therefore, \(\frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} * (\frac{\sqrt{3}}{\sqrt{3}}) = \frac{\sqrt{3}}{3}\).

Correct choice is (b) 2

The explanation is: We know that \((\SQRT{a}+\sqrt{b}) * (\sqrt{a}-\sqrt{b})\) = a-b

By applying that RULE here, we GET = \((\sqrt{7}+\sqrt{5}) * (\sqrt{7}-\sqrt{5})\)

= 7-5

= 2.

The CORRECT ANSWER is (c) rational

Easy explanation: π – π = 0

Which is rational number SINCE it is terminating.

Correct choice is (B) IRRATIONAL

The explanation: \(\sqrt{2}\) = 1.414213…

Since the EXPANSION of √2 is non-terminating and non-recurring, result of 3 + \(\sqrt{2}\) would be also non-terminating and non-recurring. So we can say that 3 + \(\sqrt{2}\) is an irrational number.

Correct option is (d) rational or irrational

To explain: Let’s take two irrational numbers to understand this.

Consider \(\sqrt{2}\) and –\(\sqrt{2}\)

By adding these, we get \(\sqrt{2}\) + (-\(\sqrt{2}\)) = 0

Which is rational number SINCE it is terminating.

Now, consider \(\sqrt{5}\) and \(\sqrt{2}\)

By adding these, we get \(\sqrt{5}\) + \(\sqrt{2}\)

Which is irrational since the EXPANSION of √5 and √2 is non-terminating and non-recurring.

Now, let’s take two numbers for subtraction.

Consider \(\sqrt{2}\) and \(\sqrt{2}\).

By subtracting these, we get \(\sqrt{2} – \sqrt{2}\) = 0

Which is rational number since it is terminating.

Consider \(\sqrt{5}\) and \(\sqrt{2}\)

By subtracting these, we get \(\sqrt{5}\) – \(\sqrt{2}\)

Which is irrational since the expansion of \(\sqrt{5}\) and \(\sqrt{2}\) is non-terminating and non-recurring.

Hence, we can say that if we ADD or subtract two irrational numbers, we get rational or irrational number.

The correct choice is (c) IRRATIONAL

Easiest explanation: LET’s take TWO non-zero numbers to understand this. One rational and one irrational number.

Consider two numbers, 3 and \(\sqrt{2}\)

Adding these two, we get 3 + \(\sqrt{2}\)

Now, \(\sqrt{2}\) = 1.414213…

Expansion of √2 is non-terminating, hence result of 3 + \(\sqrt{2}\) would be non-terminating and non-recurring.

The same will happen while SUBTRACTING this.

Hence, we can say that if we add or subtract an irrational number and a rational number (non-zero) then we get an irrational number.

Correct CHOICE is (c) rational Number

Easy explanation: Let’s take two non-zero rational numbers to understand this.

Consider two numbers, 1/2 and 3/4

Adding these two, we get 1/2 + 3/4 = {4 + (2*3)}/2*4

= 10/8

= 5/4

Which is rational number since its RESULT is terminating (5/4 = 1.25).

Subtracting these two, we get 3/4 – 1/2 = {(2*3) – 4}/2*4

= 2/8

= 1/4

Which is rational number since its result is terminating (1/4 = 0.25).

HENCE, result of two terminating numbers would ALSO be terminating so we can say that Summation or Subtraction of two rational number is always rational number.

However, as we saw in this example, it is not necessary that the result of summation or subtraction of two non-zero rational number would be Integer. So the result may or may not be Natural number or Whole number.

The correct answer is (d) irrational NUMBER

The explanation is: √5=2.236067…

This is not an integer, so it doesn’t belong to natural and whole number SET by DEFINITION. MOREOVER, expansion of √5 is non-terminating and non-recurring. Hence, it is an irrational number.

Right OPTION is (B) 0.343403400…

Explanation: 1/3=0.3333… and 2/3=0.6666…0.333… and 0.555… are non-terminating and recurring, so they can be REPRESENTED in the form p/Q where p and q are integers. Hence they are rational numbers. 0.343403400 and 0.67670671… are non-terminating and non-recurring. Hence they are irrational numbers. But, among 0.343403400…and 0.67670671, only 0.343403400 is between 1/3 and 2/3. So option 0.343403400… is CORRECT.

The correct ANSWER is (a) True

For EXPLANATION: LET’s take any TWO numbers to UNDERSTAND this.

For example, 3/4 and 4/5

We know that 3/4=0.75 and 4/5=0.8

For a number to be irrational, it has to be non-terminating and non-recurring.

We can find infinite numbers between 0.75 and 0.8 which are non-terminating and non-recurring.

Like 0.756757758…., 0.78754729… and so on.

The CORRECT option is (b) irrational number

Explanation: Given number is not an INTEGER, so it does not belong to natural number and whole number set.

It is not-terminating and non-recurring, so we can SAY that 0.238765563246…is an irrational number and not a RATIONAL number.

Correct option is (a) True

The best I can explain: Let’s take a known rational number to understand this.

For example, 3/4

We know that 3/4 is a rational number because both 3 and 4 are natural

numbers and 3/4=0.75 which is TERMINATING expansion.

Now, let’s take ONE example of non-terminating and recurring expansion.

For example 0.5787878…

Let x=0.5787878…

Then 100x=57.878787…

100x=57.3 + 0.5787878…

100x=57.3 + x

99x=57.3

Then, x=57.3/99

x=573/990

This is of the form of p/q where p and q are natural numbers. Hence we can SAY that .5787878… is rational number.

Hence, we can conclude that the DECIMAL expansion of rational numbers is either terminating or non-terminating and recurring (REPEATING).

We can also conclude that ‘The decimal expansion of irrational numbers is non-terminating and non-recurring.’

Right option is (B) 5.4121212…

The best I can explain: Bar SIGN is USED to SHOW the blocks of numbers which are repeated. Here, the sign of ‘bar’ ( ̅ )is only over two numbers, 1 and 2 which means block of these two numbers is repeated INFINITE times. There is no sign of bar over ‘4’ which shows that ‘4’ is not repeated.

Right answer is (a) True

To explain I would say: ACCORDING to definition, Real numbers comprise rational and irrational numbers. So we can say that every POINT on number LINE REPRESENTS a unique real number and vice versa.

Correct answer is (a) \(\frac{\sqrt{9}}{2}\)

EASY EXPLANATION: \(\frac{\sqrt{9}}{2} = \frac{3}{2}\) where 3 and 2 are integers. So \(\frac{3}{2}\) is RATIONAL number by definition.

π ≈ 3.14, \(\sqrt{3}\) ≈ 1.73 \(\sqrt{11}\) ≈ 3.31

As we can SEE that none of the above three numbers is integer so they are not rational number by definition. (Rational number = p/q where p and q are integers and q ≠ 0).

The correct choice is (c) INFINITE

Explanation: There are infinite numbers between 3 and 8 like 3.45, 5.56, 5.95, 6, 7.41…. and so on. All these numbers are real according to the definition of real numbers

{Real numbers} = {Rational numbers} + {IRRATIONAL numbers}

Hence we can SAY that there are infinite real numbers between any TWO integers.

Correct CHOICE is (a) True

The best I can explain: REAL NUMBERS comprise rational numbers and IRRATIONAL numbers.

{Real numbers} = {Rational numbers} + {Irrational numbers}

Hence we can SAY that all irrational numbers are real numbers.

The correct option is (a) R

To explain: As per the CONVENTIONAL notation, IRRATIONAL numbers are denoted by ‘R’.

W, Q and N are USED for Whole numbers, Rational numbers and Natural numbers RESPECTIVELY.

Correct answer is (b) irrational NUMBER

For explanation I would say: ACCORDING to definition, a number is said to be irrational if it can’t be represented in the form of p/q, where p and q are integers and q ≠ 0.

Here, \(\sqrt{3}\) ≈ 1.73 which is not an integer. So as PER the definition, the given number is irrational number.

The CORRECT answer is (d) S

To elaborate: As per the CONVENTIONAL notation, irrational numbers are denoted by ‘S’.

N, Z and Q are used for NATURAL numbers, Integers and Rational numbers respectively.

Right option is (C) Q

The best I can EXPLAIN: Rational NUMBERS are DENOTED by Q. Q = p/q where p and q are integers and q ≠ 0.

The correct answer is (a) TRUE

The explanation: N = {1, 2, 3….}

W = {0, 1, 2, 3….}

Z = {…-3, -2, -1, 0, 1, 2, 3….}

Q = p/q where p and q are INTEGERS and q ≠ 0

We can SEE that all the members of N are contained in W, Z and Q hence, it is true that N is subset of W, Z and Q.

Correct choice is (a) FOUR

For explanation: W = {0, 1, 2, 3…….}. So whole NUMBERS between -3 and 3 are 0, 1, 2 and 3.

Hence, there are four numbers between -3 and 3 that are whole numbers.

The CORRECT answer is (d) √2

To explain: A number is SAID to be RATIONAL number if it can be represented by p/q where p and q are integers and q≠0 but √2 = √2/1 and √2 = 1.414213… which is not integer HENCE, √2 is not rational number.

Correct choice is (c) infinite

The best I can explain: Between 2 and 3, there are infinite numbers which are RATIONAL numbers like 11/5, 12/5, 13/6, 17/8 and so on. They all are integers and no DENOMINATOR is zero so they are rational numbers. Thus, we can find infinite numbers who satisfies this condition.

So we can SAY that “There are infinite rational numbers between any TWO integers”.

Right answer is (a) Q = P/q where p and q are integers and q=0

The best explanation: Q = p/q where p and q are integers and q = 0 is an incorrect option. In the option, q=0 is GIVEN which is incorrect it should be q ≠ 0. All other options are correct since N set includes all positive integers, W set includes all integers and ZERO and Z set includes zero and all positive and negative integers.

The correct choice is (d) Z

Easiest explanation: Integers are denoted by Z where Z CONTAINS {…-2, -1, 0, 1, 2 …}.