✅ Radical Formula ⭐️⭐️⭐️⭐️⭐️

Radical

In maths, a radical is the opposite of an exponent that is represented with a symbol ‘√’ also known as root. It can either be a square root or a cube root and the number before the symbol or radical is considered to be an index number or degree. This number is a whole number represented as an exponent that cancels out the radical. Let us learn more about the concept of radicals, the formula, and solve a few examples.

What is Radical?

The radical of a number is the same as the root of a number. The root can be a square root, cube root, or in general, n^th root. Thus, any number or expression that uses a root is known as a radical. The term radical is derived from the Latin word Radix which means root. The radical can describe different kinds of roots for a number such as square root, cube root, fourth root, and so on. The number written before the radical is known as the index number or degree. This number helps in telling us how many times the number would be multiplied by itself to equal the radicand. This is considered to be the opposite of an exponent just like addition being the opposite of subtraction and division being the opposite of multiplication. For example: ∛125 = 5 as 5 × 5 × 5 = 125.

Radical Definition

The symbol ‘√’ that expresses a root of a number is known as radical and is read as x radical n or n^th root of x. The horizontal line covering the number is called the vinculum and the number under it is called the radicand. The number n written before the radical is called the index or degree. Some examples of radicals are √7, √2y+1, etc.

A radical can also be associated with the following terms:

• An equation that is inside a radical is known as a radical equation.
• An expression that lies inside a square root is known as a radical expression.
• An inequation that is inside a radical is known as radical inequality.

Radical General Rules

Mentioned below are a few general rules for a radical.

• If the number is positive under the radical, the result will be positive.
• If the number is negative under the radical, the result will be negative.
• If the number under the radical is negative and an index is an even number, the result will be an irrational number.
• If an index is not mentioned, the radical will be square root.
• Multiplication of numbers under the same radical and index is possible. For example, ∛12 × ∛10 = ∛120.
• Division is possible for numbers under the same radical. For example, √8/√4 = √8/4 = √2.
• The reverse of the multiplication rule is possible, where the number is split under the same radical. For example, √27 = √9 × √3 = ∛3 × √3.
• The radical can be written in its exponent form as well in any equation. For example, √x = 25 (√x)² = (25)² x = 5.
• The inverse exponent of the index number is equivalent to the radical itself. For example, √7 = (7)^1/2.

Radical Formula

To solve a radical equation, it has to be made radical-free. To make an equation of n^th root radical free, we power both sides of the equation with ‘n’. This masked the radical equation-free from radical. Let’s look into the radical formula below.

n√ x=p

x^1/n = p

(x^1/n)ⁿ = pⁿ

x = pⁿ

where,

• The n√ symbol is known as the radical of n^th root.
• ‘n’ is known as the index.
• The expression or variable inside the radical symbol i.e, x is known as the radicand.

Let’s take some examples to understand the radical formula.

Radical Examples

Example 2: Solve the radical: $\sqrt[3]{3x}$

= 8 using the radical formula.

Solution:

Given:

$\sqrt[3]{3x}$

= 8

To make it radical free, using radical formula

x^1/3 = 8

(x^1/3)³ = 8³

x = 8³

x = 512

Therefore, Thus, the value of x for $\sqrt[3]{3x}$

= 8 is 512.

A radical equation is an equation in which a variable is under a radical. To solve a radical equation:

1. Isolate the radical expression involving the variable. If more than one radical expression involves the variable, then isolate one of them.

2. Raise both sides of the equation to the index of the radical.

3. If there is still a radical equation, repeat steps 1 and 2; otherwise, solve the resulting equation and check the answer in the original equation.

By raising both sides of an equation to a power, some solutions may have been introduced that do not make the original equation true. These solutions are called extraneous solutions.

Example 1

 

Important Note: The step when we square both sides can introduce some ” extraneous solutions “. You must check for validity of solutions.

Substituting x=21

:

So the second solution doesn’t check out. Therefore x=1 is the extraneous solution. The only valid solution is x=21

General rules of a Radical Formula

The following are some general guidelines for radicals.

• The resultant will be positive if there is a positive number under the radical. 
• The resultant will be negative if the number under the radical is negative. Take note radical negative numbers are only calculated if their index is odd.
• If the index of the radical is not mentioned, then it is considered a square root.
• Multiplication of numbers applies to numbers that share the same radical and index. For example, 5√21 × 5√15 =  5√(21×15) =  5√315.
• Similarly, the division is also applicable to numbers that have the same radical. For example, ³√25/³√5 = ³√(25/5) = ³√5.
• The reverse of the multiplication can be applied by splitting the number under the same radical. For example, √32 = √16 × √2 = 4√2.
• In any equation, a radical can be expressed in its exponential form.
• An index number’s inverse exponent is equivalent to the radical itself.

Root of a product	n√(a × b) =n√(a) × n√(b)

Root of a quotient	n√(a/b) = n√(a)/ n√(b)
Fractional Exponent	n√(a)m = (a)m/n

ⁿ√(x) is a radical expression of the “nth root of x”. A radical expression is said to be simplified, it has to be radically free. So, for making the given expression radical free, we need to power both sides of the given equation with “n”. 

ⁿ√(x) = p

(x)^1/n = p

(x^1/n)ⁿ = (p)ⁿ

x = (p)ⁿ

Where x is the radicand,

n is the index of the radical, and

 

(ⁿ√) is the radical symbol or nth root.

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 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: Simplify ³√36a⁵b² ³√6ab.

Solution:

Given,

³√36a⁵b² ³√6ab

= ³√(36a5b2) × (6ab)

=  ³√(216a⁶b³)

= ³√(6³a⁶b³)

= 6a²b

Hence, the value of the given radical expression is 6a²b.

Problem 6: 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).

We summarize the steps here. We have adjusted our previous steps to include more than one radical in the equation This procedure will now work for any radical equations.

Be careful as you square binomials in the next example. Remember the pattern is

Use Radicals in Applications

As you progress through your college courses, you’ll encounter formulas that include radicals in many disciplines. We will modify our Problem Solving Strategy for Geometry Applications slightly to give us a plan for solving applications with formulas from any discipline.

One application of radicals has to do with the effect of gravity on falling objects. The formula allows us to determine how long it will take a fallen object to hit the gound.

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes.

Key Concepts

• Binomial Squares

• Solve a Radical Equation
  1. Isolate one of the radical terms on one side of the equation.
  2. Raise both sides of the equation to the power of the index.
  3. Are there any more radicals?
      

     If yes, repeat Step 1 and Step 2 again.

      

     If no, solve the new equation.

  4. Check the answer in the original equation.

• Problem Solving Strategy for Applications with Formulas
  1. Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.
  2. Identify what we are looking for.
  3. Name what we are looking for by choosing a variable to represent it.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.

Falling Objects

• On Earth, if an object is dropped from a height of h feet, the time in seconds it will take to reach the ground is found by using the formula

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