Solved Examples on Square Roots

Example 1: Estimate the square root of 72.

Solution:

Perfect squares closest to 72 are 64 and 81.

Square root of 64 is 8, and the square root of 81 is 9.

Therefore, the square root of 72 is estimated to be between 8 and 9.

Example 2: Simplify √27.

Solution:

We can factor 27 as √(9 × 3), and since the square root of 9 is 3, we can simplify it as 3√3.

Example 3: Simplify √75.

Solution:

We can factor 75 as √(25 × 3), and since the square root of 25 is 5, we can simplify it as 5√3.

Example 4: Simplify 4 / (√2 + √3)

Solution:

To rationalize the denominator, we multiply both the numerator and denominator by (√2 – √3).

= 4×(√2 – √3)/(√2 + √3)(√2 – √3)

= 4×(√2 – √3)/(√2x√2 – √3 √3)

= 4×(√2 – √3)/(2-3)

This gives us [4(√2 – √3)] / (-1), which simplifies to -4(√2 – √3)

Example 5: Simplify (3 + √5) / (√5 – 1)

Solution:

To rationalize the denominator, we multiply both the numerator and denominator by (√5 + 1).

= (3 + √5)(√5 + 1) / (√5 – 1)(√5 + 1) (multiplying by the conjugate of the denominator)

= (3√5 + 3 + √5√5 + √5) / (5 – 1) (expanding the numerator and denominator)

= (4√5 + 8) / 4

= 4(2 + √5) / 4 (cancelling numerator and denominator)

= 2+√5

This gives us [(3 + √5)(√5 + 1)] / (5 – 1), which simplifies to 2 + √5

Example 6: Find the square root of -16.

Solution:

Since the square root of -16 is not a real number,

We can represent it as a complex number of the form a + bi. In this case, we have a = 0 and b = 4.

Therefore, the square root of

-16 = √(i²(4)²)

= 4i

Example 7: Find the square root of -3 – 4i.

Solution:

To find the square root of a complex number we can use the formula,

√(a + bi) = ±(√[(a + √(a² + b²))/2] + i√[(|a – √(a² + b²)|)/2])

Applying this formula to the complex number -3 – 4i, we have a = -3 and b = -4. Therefore, we can substitute these values into the formula,

√(-3 – 4i) = ±(√[(-3 + √(9 + 16))/2] + i√[(|-3 – √(9 + 16)|)/2])

= ±(√[(-3 + √(25))/2] + i√[(|-3 – √(25)|)/2])

= ±(√[(-3 + 5)/2] + i√[(|-3 – 5|)/2])

= ±(√(2/2) + i√(|-8|/2))

= ±(√(2/2) + i√(8/2))

= ±(√1 + i√4)

= ±(1 + 2i)

Example 8: Simplify 4 / (√2 – √3)

Solution:

To rationalize the denominator, we multiply both the numerator and denominator by (√2 + √3).

= 4 × (√2 + √3)/(√2 – √3)(√2 + √3)

= 4 × (√2 + √3)/(√2x√2 – √3 √3)

= 4 × (√2 + √3)/(2-3)

This gives us [4(√2 + √3)] / (-1), which simplifies to -4(√2 + √3)

Square root symbol or square root sign is denoted by the symbol ‘√’. It is a mathematical symbol used to represent square roots in mathematics. The square root symbol (√) is also called Radical. For example, we write the square root of 4 as √(4). It is read as root 4 or the square root of 4.

Let’s learn about square root, its representation, simplification, and others in this article.