Introduction to rules of simplifying radicals:
Radical symbol used to indicate the square root or nth root. Radical is an algebraic group, a concept in algebraic group theory. A branch of mathematics, a radical of a ring is an ideal of elements of the ring. Radical of a module, in the theory of modules, the radical of a module is a component in the theory of structure and classification. The radical sign is v . (source: wikipedia)
Rules of Simplifying Radicals:
Radical is a perfect and significant idea in abstract algebra. The cubic root of x know how to be expressed as `root(5)(x)`
`root(n)(ab)` = `root(n)(a)` `root(n)(b)`
`root(n)((a/b))` = `(root(n)(a))` / `(root(n)(b))`
`root(n)(x)` = (x)(1/n) and `root(n)(x^n)` = x(n/n) = x
`root(n)(a)`m =( `root(n)(a)` )m = (a1/n )m = am/n
` sqrt(-1)` × `sqrt(-1)` = -1 where as `sqrt((-1)*(-1))` = 1
Examples for Rules of Simplifying Radicals:
Example 1:
Simplifying the rules of given radical expression: `sqrt((9x^2)/(y^5z^5)) `
Solution:
Step 1: Multiplying and divide by yz
Step 2: ` sqrt((9x^2)/(y^5z^5))` = `sqrt((9x^2 * yz) / (y^5z^5 *yz))`
Step 3: Multiply the variable with exponent
= `sqrt((9x^2yz) / (y^6z^6))`
Step 4: Square root of x2y6z6 = xy3z3
Step 5: Square root of 9 = `sqrt(3 * 3) ` = 3
=` ((3x) / (y^3z^3)) sqrt(yz)`
so the answer is ` ((3x) / (y^3z^3)) sqrt(yz)`
Example 2:
Simplifying the rules of radical expression: `sqrt(b^4/a^7) ` + `sqrt(a)` = `sqrt(a)` ` ((b+a^3)/a^3)`
Solution:
Step 1: Multiplies and divides by a in first term
Step 2: `sqrt(b^4/a^7)` = `sqrt ((b^4* a) / (a^7 *a))`
Step 3: Multiply the variable with exponent
= `sqrt ((b^4 * a) / (a^8))`
Step 4: Square root of b4 a = b2 `sqrt(a)`
Step 5: Square root of a8 = a4
`sqrt ((b^4 * a) / (a^8))` = ` b^2/a^4 sqrt(a)`
Step 6: Adding the both term `sqrt(b^4/a^7) ` + `sqrt(a)` =` b^2/a^4 sqrt(a)` + `sqrt(a)`
= `sqrt(a)` ( `b^2/a^4 ` +1 )
Step 7: = `sqrt(a)` ` ((b^2+a^4)/a^4)`
Hence the given radical expression has been proved
Example 3:
Simplifying the rules of the given radical expression: `sqrt(5x^3) (sqrt(9x^3))`
Solution:
Step 1: the given radical expression is
= `sqrt(5x^3) (sqrt(9x^3))`
Step 2: `sqrt(5x^3) (sqrt(9x^3))`
=` (xsqrt(5x) (3)xsqrtx)`
Step 3: `3x^2 sqrt(5 * x * x) `
Step 4: 3x2x `sqrt(5)`
= 3 x3`sqrt5`
so the answer is 3 x3`sqrt5`
Example 4:
Simplifying the rules of the given radical expression : `(root3 16)^3 * (sqrt49)^2`
Solution:
Step 1: Given radicals ` (root3 16)^3 * (sqrt49)^2`
Step 2: = `(root3 16)^3 * (sqrt49)^2`
Step 3: Radical rules is ( `root(n)(a)` )m = am/n
Step 4: = 163/3 * 492/2
161 * 491
Step 5: 16 * 49
Step 6: so the answer is 784
Radical symbol used to indicate the square root or nth root. Radical is an algebraic group, a concept in algebraic group theory. A branch of mathematics, a radical of a ring is an ideal of elements of the ring. Radical of a module, in the theory of modules, the radical of a module is a component in the theory of structure and classification. The radical sign is v . (source: wikipedia)
Rules of Simplifying Radicals:
Radical is a perfect and significant idea in abstract algebra. The cubic root of x know how to be expressed as `root(5)(x)`
`root(n)(ab)` = `root(n)(a)` `root(n)(b)`
`root(n)((a/b))` = `(root(n)(a))` / `(root(n)(b))`
`root(n)(x)` = (x)(1/n) and `root(n)(x^n)` = x(n/n) = x
`root(n)(a)`m =( `root(n)(a)` )m = (a1/n )m = am/n
` sqrt(-1)` × `sqrt(-1)` = -1 where as `sqrt((-1)*(-1))` = 1
Examples for Rules of Simplifying Radicals:
Example 1:
Simplifying the rules of given radical expression: `sqrt((9x^2)/(y^5z^5)) `
Solution:
Step 1: Multiplying and divide by yz
Step 2: ` sqrt((9x^2)/(y^5z^5))` = `sqrt((9x^2 * yz) / (y^5z^5 *yz))`
Step 3: Multiply the variable with exponent
= `sqrt((9x^2yz) / (y^6z^6))`
Step 4: Square root of x2y6z6 = xy3z3
Step 5: Square root of 9 = `sqrt(3 * 3) ` = 3
=` ((3x) / (y^3z^3)) sqrt(yz)`
so the answer is ` ((3x) / (y^3z^3)) sqrt(yz)`
Example 2:
Simplifying the rules of radical expression: `sqrt(b^4/a^7) ` + `sqrt(a)` = `sqrt(a)` ` ((b+a^3)/a^3)`
Solution:
Step 1: Multiplies and divides by a in first term
Step 2: `sqrt(b^4/a^7)` = `sqrt ((b^4* a) / (a^7 *a))`
Step 3: Multiply the variable with exponent
= `sqrt ((b^4 * a) / (a^8))`
Step 4: Square root of b4 a = b2 `sqrt(a)`
Step 5: Square root of a8 = a4
`sqrt ((b^4 * a) / (a^8))` = ` b^2/a^4 sqrt(a)`
Step 6: Adding the both term `sqrt(b^4/a^7) ` + `sqrt(a)` =` b^2/a^4 sqrt(a)` + `sqrt(a)`
= `sqrt(a)` ( `b^2/a^4 ` +1 )
Step 7: = `sqrt(a)` ` ((b^2+a^4)/a^4)`
Hence the given radical expression has been proved
Example 3:
Simplifying the rules of the given radical expression: `sqrt(5x^3) (sqrt(9x^3))`
Solution:
Step 1: the given radical expression is
= `sqrt(5x^3) (sqrt(9x^3))`
Step 2: `sqrt(5x^3) (sqrt(9x^3))`
=` (xsqrt(5x) (3)xsqrtx)`
Step 3: `3x^2 sqrt(5 * x * x) `
Step 4: 3x2x `sqrt(5)`
= 3 x3`sqrt5`
so the answer is 3 x3`sqrt5`
Example 4:
Simplifying the rules of the given radical expression : `(root3 16)^3 * (sqrt49)^2`
Solution:
Step 1: Given radicals ` (root3 16)^3 * (sqrt49)^2`
Step 2: = `(root3 16)^3 * (sqrt49)^2`
Step 3: Radical rules is ( `root(n)(a)` )m = am/n
Step 4: = 163/3 * 492/2
161 * 491
Step 5: 16 * 49
Step 6: so the answer is 784
