How do you know if a radical is rational or irrational? (original) (raw)

Last Updated : 23 Jul, 2025

Radical is rational only when the square root of any number is itself a number in the result or if the number is the perfect square of radical then it's a rational number otherwise it's an irrational number.

**For example:

√25 = Square root of 25 is 5,

Which is a perfect square of 5. Hence 5 can be represented in the form of p/q,

Therefore √25 is a rational radical.

√15 = Square root of 15 is 3.87298334...

Which is not a perfect square, hence it cannot be represented as p/q, and neither is it terminating nor recurring after decimal.

Therefore √15 is an irrational radical.

Types of Numbers Involved

Real Numbers

Rational numbers, such as positive and negative integers, fractions, and irrational numbers, are all examples of Real numbers. The set of real numbers, indicated by R, is the union of the set of rational numbers (Q) with the set of irrational numbers. This means that real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. For example, 3, 0, 1.5, 3/2, 5, and so on are all real numbers.

Rational number*

Any integer that can be expressed as a fraction p/q is called a rational number. In a fraction, the numerator is 'p,' and the denominator is 'q,' where 'q' is not equal to zero. A natural number, a whole number, a decimal, or an integer are all examples of rational numbers.

1/2, -2/3, 0.5, and 0.333, for example, are rational numbers.

Irrational numbers*

Irrational numbers are a set of real numbers that cannot be represented as a fraction p/q, where p and q are integers and the numerator q is not equal to zero (q ≠0).

Irrational numbers, such as (pi), are one example. 3.14159265...

The decimal value in this case is never finished. As a result, irrational numbers include numbers like 2, -7, and so on.

Radical*

The radical and root of a number are the same thing. The root can be a square root, a cube root, or an nth root in general. As a result, a radical is any number or expression that uses a root. The radical may be used to explain several types of roots for a number, such as square, cube, fourth, and so on. The index number or degree is the number written before the radical. This number indicates how many times the radicand must be multiplied by itself to equal the number.

The symbol '√' for a number's root is known as radical, and it is written as x radical n or nth root of x.

Sample Questions

**Question 1: How do you know √16 is a rational or not?

**Solution:

Given, √16

Here the square root of 16 is 4.

Which shows it is a perfect square, and 4 can be represented in form of 4/1.

Therefore √16 is a rational radical.

**Question 2: Examine whether Radical of 8 is rational or not?

**Solution:

Radical is rational only when the square root of any number is itself a number in result or if the number is the perfect square of radical then its a rational number otherwise its an irrational number.

Here given: √8

Square root of 8 is 2.828427.. which is not a perfect square.

Therefore radical 8 is not a rational number.

**Question 3: Examine whether Radical of 100 is rational or not?

**Solution:

Radical is rational only when the square root of any number is itself a number in result or if the number is the perfect square of radical then its a rational number otherwise its an irrational number.

Here given: √100

Square root of 100 is 10. which is a perfect square.

Therefore radical 100 is a rational number.

**Question 4: Examine whether the Radical of 5 is rational or not?

**Solution:

Radical is rational only when the square root of any number is itself a number in result or if the number is the perfect square of radical then its a rational number otherwise its an irrational number.

Here given: √5

Square root of 5 is 2.236067.. which is not a perfect square.

Therefore radical 5 is an irrational number.

**Question 5: Examine whether Radical of 144 is rational or not?

**Solution:

Radical is rational only when the square root of any number is itself a number in result or if the number is the perfect square of radical then its a rational number otherwise its an irrational number.

Here given: √144

Square root of 144 is 12, which is a perfect square.

Therefore radical 144 is a rational number.

**Question 6: Examine whether Radical of 133 is rational or not?

**Solution:

Radical is rational only when the square root of any number is itself a number in result or if the number is the perfect square of radical then its a rational number otherwise its an irrational number.

Here given: √133

Square root of 133 is 11.53256.. , which is a perfect square.

Therefore radical of 133 is not a rational number.

**Also read,

Similar Questions

How can you determine if a square root is a rational number?

**Perfect Square Check : If the number under the square root is a perfect square (i.e., it can be expressed as n², where n is an integer), then the square root is rational. For example, √16 = 4² , which is rational because 16 is a perfect square (4² ).

**Non-Perfect Square : If the number is not a perfect square, then the square root is irrational. For example, √7 is approximately 2.6457513110645906.Thus √7 is irrational because 7 is not a perfect square.

How do you identify whether the nth root of a number is rational or irrational?

**Perfect nth Power: If the number is a perfect nth power of an integer (i.e., it can be expressed as n^(k) , where k is is a positive integer and n is an integer), then the nth root is rational. For example, ∛27 = 3, which is rational because 27 is 3³.

**Non-Perfect nth Power: If the number is not a perfect nth power, then the nth root is irrational. For example, ∜20 is irrational because 20 is not a perfect fourth power of any positive integer.

How can you tell if a number is a perfect square or a perfect cube?

**Perfect Square: A number is a perfect square if it can be expressed as n² where n is an integer. For example, 25 is a perfect square because 5² = 25. To check, you can take the square root of the number and see if it is an integer.

**Perfect Cube: A number is a perfect cube if it can be expressed as n³ where n is an integer. For example, 27 is a perfect cube because 3³ = 27. To check, you can take the cube root of the number and see if it is an integer.

What methods can be used to prove the irrationality of a number?

Contradiction Method: Assume that the number is rational and express it as a fraction p/q in simplest form. Show that this assumption leads to a contradiction. For example, to prove √2 is irrational, assume √2 = p/q, then show that both p and q must be even, contradicting the assumption that p/q is in simplest form.

Prime Factorization: Show that the number cannot be expressed as a finite or repeating decimal or a fraction. For example, proving that 𝜋 and 𝑒 are irrational involves advanced methods beyond simple factorization, but for radicals, factorization can be useful.

How do you find the decimal representation of a radical?

Approximation: Use a calculator or computational tool to approximate the decimal value. For example, √7 is approximately 2.6457513110645906.

Manual Calculation: Use methods like long division or numerical approximation techniques to estimate the value to a desired precision.