The Simplest Transformations of Arithmetic Root (Radical)

When we transform the arithmetic root we should use their properties.

Example 1. Simplify the following: `sqrt(45a^5)`.

Using property 1, we will obtain `sqrt(45a^5)=sqrt(9a^4*5a)=sqrt(9)*sqrt(a^4)*sqrt(5a)=3a^2sqrt(5a)`.

Such transformation is called factoring out of root.

Example 2. Simplify the following: `(root(3)(a^2))^5`.

Using property 3, we will obtain `(root(3)(a^2))^5=root(3)((a^2)^5)=root(3)(a^10)`. Let′s simplify radical expression, for this we will factor out of root.

Then `root(3)(a^10)=root(3)(a^9*a)=root(3)(a^9)*root(3)(a)=a^3root(3)(a)`.

Example 3. Simplify the following: `root(4)(x^2root(3)(x))`.

Let′s transform expression `x^2root(3)(x)`, for this we will factor in of root:

`x^2root(3)(x)=root(3)((x^2)^3)*root(3)(x)=root(3)(x^6)*root(3)(x)=root(3)(x^6*x)=root(3)(x^7)`.

According to property 4, we have `root(4)root(3)(x^7)=root(12)(x^7)`.

Example 4. Simplify the following: `root(30)(2^9)` .

According to property 5, we can divide the index of radical and exponent of radicand by the same natural number. Dividing these indicators by 3, we will obtain `root(30)(2^9)=root(10)(2^3)=root(10)(8)`.

Example 5. Simplify the following: `root(5)(a)*root(5)(a^2)`.

According to property 1, if we need to multiply the same degree we should multiply the radicands and extract the root of the same degree from obtained result. So, `root(5)(a)*root(5)(a^2)=root(5)(a*a^2)=root(5)(a^3)`.

Example 6. Simplify the following: `root(3)(a)*root(6)(a)`.

At first we should reduce the radicals to the same index. According to property 5, we can divide the index of radical and exponent of radicand by the same natural number. That is why `root(3)(a)=root(6)(a^2)`. Then we have `root(6)(a^2)xxroot(6)(a)=root(6)(a^3)`. Dividing the index of radical and exponent of radicand by 3, we will obtain `root(6)(a^3)=sqrt(a)`.

When we perform the operations with radicals, we often transform into fractional exponents. For example,

`root(8)(x^3)*root(12)(x^7)=x^(3/8)*x^(7/12)=x^(3/8+7/12)=x^(23/24)=root(24)(x^23)`.