Radical Formula
In mathematics, an expression with a root is referred to as a radical. A radical can be used to describe various types of roots of a number, including square roots, cube roots, fourth roots, etc. For example, common radicals like the square root and cube root are expressed by the symbols β and Β³β, respectively, where β3β is the degree or index of the number. If the index of the radical is not mentioned, then it is considered a square root. The word βRadicalβ is extracted from the Latin word βradixβ which means βRootβ.
Table of Content
- What is Radical?
- General Rules of a Radical Formula
- Solution of Radical Equation
- Sample Problems on Radical Formula
- FAQs on Radical
What is Radical?
Radical in mathematics are similar to the roots of the number. Any numbers expressed as root, nth root, or others are called radicals. For example: β(12), For example: β(5), (11)1/5, etc. all are radicals. Let us consider an example to understand the concept of radical better.
In the figure shown above, βnβ is the index of the radical, β(a + 3b)β is the radicand, and β(nβ)β is the radical symbol, and it is symbolically written as βnth root of (a+3b).β The index of a radical helps to determine how many times a number must be multiplied by itself to equal a radicand. And also an index of a radical is regarded as an antithesis of an exponent.
General Rules of a Radical Formula
The following are some general rules used 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, 5β21 Γ 5β15 = 5β(21Γ15) = 5β315.
- Similarly, the division is also applicable to numbers that have the same radical. For example, Β³β25/Β³β5 = Β³β(25/5) = Β³β5.
- The reverse of the multiplication can be applied by splitting the number under the same radical. For example, β32 = β16 Γ β2 = 4β2.
- In any equation, a radical can be expressed in its exponential form.
- An index numberβs inverse exponent is equivalent to the radical itself.
Root of a product | nβ(a Γ b) =nβ(a) Γ nβ(b) |
---|---|
Root of a quotient | nβ(a/b) = nβ(a)/ nβ(b) |
Fractional Exponent | nβ(a)m = (a)m/n |
Solution of Radical Equation
nβ(x) is a radical expression of the βnth root of xβ. 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 βnβ.
nβ(x) = p
(x)1/n = p
(x1/n)n = (p)n
x = (p)n
where,
- x is Radicand
- n is Index of Radical
- (nβ) is Radical Symbol or nth root
Articles Related To Radical Formula:
Sample Problems on Radical Formula
Problem 1: Solve the radical, βy = 11, using the radical formula.
Solution:
Given,
βy = 11
To make the given expression radical-free, use the radical formula.
(y)1/2 = 11
Now squaring on both sides we get
β [(y)1/2]2 = (11)2
β y = (11)2 β y = 121
Hence, the value of y is 121.
Problem 2: Solve the radical expression (7 + 5βa)/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 + 5βa)/b
= (7 + 5β36)/4
= (7 + 5 Γ 6)/4
= 37/4 = 9.25
Hence, the value of the given radical expression is 9.25.
Problem 3: Simplify β(175a4b5)/β(7b).
Solution:
β(175a4b5)/β(7b)
By using the quotient rule, we get
=[Tex] \sqrt{\frac{175a^{4}b^{5}}{7b}} [/Tex]
= β(25a4b4)
= 5a2b2
Hence, the value of the given radical expression is 5a2b2.
Problem 4: Solve β(3x+9) β 6 = 0
Solution:
Given,
β(3x+9) β 6 = 0
β β(3x+9) = 6
Now squaring on both sides we get
β (3x + 9) = (6)2
β 3x + 9 = 36
β 3x = 36 β 9 = 27
β x = 27/3 = 9
Hence, the value of x is 9.
Problem 5: Simplify Β³β36a5b2 Β³β6ab.
Solution:
Given,
Β³β36a5b2 Β³β6ab
= Β³β(36a5b2) Γ (6ab)
= Β³β(216a6b3)
= Β³β(63a6b3)
= 6a2b
Hence, the value of the given radical expression is 6a2b.
Problem 6: Find the value of 3/(2+β5).
Solution:
Given,
Now, multiply and divide the given term with (2 β β5)
= 3/(2 + β5) Γ (2 β β5)/(2- β5)
= 3(2 β β5)/(22β 5) {Since, (a + b)(a β b) = a2 β b2}
= 3(2 β β5)/(4 β 5)
= 3(2 -β5)/(-1)
= 3(β5 β 2)
Hence, 3/(2 + β5) = 3(β5 β 2).
FAQs on Radical
What is the Meaning of Radical in Math?
Radical is mathematics are defined as the symbol βββ used in identifying the nth root of a number i.e. square root, cube root, etc.
What is Exponent and Radical?
Exponent defines how many times we are multiplying a number by itself. Wherea radicals used to express the root of any number.
What is the Radical Equation?
Any equation in which the variable is under radical is called the radical equation.
What is the General Formula for a Radical?
Radical expression is in general form are expressed as, nβ x. Where, βxβ is the radicand, and and βnβ is known as the index of nβ x.
How do you Solve a Radical?
Radicals can be easily solved, by Isolating one of the radical terms on one side of the equation and raise both sides of the equation to the power of the index and then simplifying accordingly.