Radical Formula

A radical is a mathematical expression that represents the nth root of a number or algebraic expression, written using the symbol √, where the value inside is called the radicand and the optional number indicating the type of root is the index.

To solve a radical equation, we remove the radical by raising both sides of the equation to the power of the index.

n√x = p
x¹⁄ⁿ = p
(x¹⁄ⁿ)ⁿ = pⁿ
x = pⁿ

where:

• n√ is the radical symbol (nth root).
• n is the index.
• x is the radicand (the value inside the radical)

General Rules

The following are some general rules used for radicals.

• If the radicand is positive, the result is positive.
• If the radicand is negative and the index is odd, the result is negative.
• If the radicand is negative and the index is even, the result is not a real number.
• If no index is specified, the radical represents a square root (index = 2).
• Radicals with the same index can be multiplied: √a × √b = √(ab).
• Radicals with the same index can be divided: √a / √b = √(a/b).
• A radical can be split into factors: √ab = √a × √b.
• A radical can be expressed in exponential form: √x = x¹⁄².
• In general, ⁿ√x = x¹⁄ⁿ.

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Sample Problems

Problem 1: Solve the radical, √y = 11, using the radical formula.

Solution:

Given,

√y = 11

To make the given expression radical-free, use the radical formula.

(y)^1/2 = 11

Now squaring on both sides we get

⇒ [(y)^1/2]² = (11)²

⇒ y = (11)² ⇒ y = 121

Hence, the value of y is 121.

Problem 2: Solve the radical expression (7 + 5√a)/b, where a = 36 and b = 4.

Solution:

Given,

a = 36 and b = 4

By substituting the values of a and b in the given radical expression we get

(7 + 5√a)/b

= (7 + 5√36)/4

= (7 + 5 × 6)/4

= 37/4 = 9.25

Hence, the value of the given radical expression is 9.25.

Problem 3: Simplify √(175a⁴b⁵)/√(7b).

Solution: 

√(175a⁴b⁵)/√(7b)

By using the quotient rule, we get

= \sqrt{\frac{175a^{4}b^{5}}{7b}}

= √(25a⁴b⁴)

= 5a²b²

Hence, the value of the given radical expression is 5a²b².

Problem 4: Solve √(3x+9) − 6 = 0

Solution:

Given,

√(3x+9) − 6 = 0

⇒ √(3x+9) = 6

Now squaring on both sides we get

⇒ (3x + 9) = (6)²

⇒ 3x + 9 = 36

⇒ 3x = 36 - 9 = 27

⇒ x = 27/3 = 9

Hence, the value of x is 9.

Problem 5: Find the value of 3/(2+√5).

Solution:

Given, 

Now, multiply and divide the given term with (2 - √5)

= 3/(2 + √5) × (2 - √5)/(2- √5)

= 3(2 - √5)/(2²- 5)     {Since, (a + b)(a - b) = a² - b²}

= 3(2 - √5)/(4 - 5)

= 3(2 -√5)/(-1)

= 3(√5 - 2)

Hence, 3/(2 + √5) = 3(√5 - 2).