Real Numbers - Class 9 Maths Free MCQ Practice Test with solutions

We know that 
By applying that rule here, we get = 
= 7-5
= 2.

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 we will get same expression since 
Therefore, 

We know that 
By applying that rule here, we get 
since 

According to laws of exponents, a^m/n = (ⁿ√a)m.
Applying that rule here, 8^2/3 = (3√8)²
= 2²
= 4.

On simplification,

√35 = √(7x5) = √7 x √5
= 8√35 = 8 x √7 x √5
= 8 x √7 x √5
     4 x √5
=2√7

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 we will get same expression since  
Therefore, 

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 we will get same expression since 
Therefore, 

According to laws of exponents, a^m/n = (n√a)m.
Applying that rule here, 5^3/8= (5³)^1/8
= (5*5*5)^1/8
= (125)^1/8.

According to laws of exponents, a^m aⁿ = a^(m+n)
Applying that rule here, (3)^1/3 * (3)^2/3 = 3^(1/3+2/3)
= 3^(1+2)/3
= 3^3/3
= 3¹ = 3.

According to laws of exponents, a^m/aⁿ = 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
= ²√5.

According to laws of exponents, a^m/aⁿ = 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.

According to laws of exponents, (a^m)*(b^m) = (ab)^m
Applying that rule here, 2³*5³=(2*5)³
=(10)³
= 1000.

To add like terms with radicals, you simply add the coefficients and keep the radical part the same.
3√6​ + 4√6​=(3 + 4)√6​ = 7√6​
Answer: (b) 7√6

Using the property of square roots 

Now, simplify 

Either you can make a fraction from two whole numbers (denominator ≠ 0), thus a rational number. Or you can't, thus irrational.

Since, 5 and 8 are the remainders of 70 and 125, respectively. Thus, after subtracting these remainders from the numbers, we have the numbers 65 = (70-5),
117 = (125 – 8), which is divisible by the required number.
Now, required number = HCF of 65,117                                     [for the largest number]
For this, 117 = 65 × 1 + 52 [∵ dividend = divisior × quotient + remainder]
⇒ 65 = 52 × 1 + 13
⇒ 52 = 13 × 4 + 0
∴ HCF = 13 
Hence, 13 is the largest number which divides 70 and 125, leaving remainders 5 amnd 8.

Let's analyze each option:

(a) x⁶−x³ is not equal to x³ because it represents a subtraction of two different powers of x.
(b) x⁶⋅x³ is equal to x⁶⁺³=x⁹, which is not equal to x³.
(c) x⁶/x³​ is equal to x⁶⁻³=x³, which is equal to x³.
(d) (x⁶)³ is equal to x^6×3=x¹⁸, which is not equal to x³.

To add like terms with radicals, you simply add the coefficients and keep the radical part the same.
4√5+5√5=(4+5)√5=9√5
Answer: (a) 9√5

Factorise the terms
5 × √5 × 3×√5 = 5×3×5 =75

Let's solve this step by step in a simple way:

We are given:

2√3 + √3

Step 1: Think of √3 as a common item.
So, we treat it like:

2√3 + 1√3

Step 2: Add the numbers in front of √3:

(2 + 1)√3 = 3√3

So, the correct answer is:

c) 3√3