Simplifying Radicals

Introduction to Radicals

A radical, or else a root, can be said to be the mathematical opposite of any specific exponent, in the similar sense that the addition can be said to be the opposite of the subtraction. The smallest radical can be said to be the square root, which is actually represented with the help of the symbol that looks like √. The next particular radical can be said to be the cube root, which is actually represented with the help of the specific symbol that actually looks like ³√.

What Are the Steps to Simplify Radicals Expressions

In order to simplify the radical expressions with the help of cube root or else the higher roots, one should certainly consider any example. One should consider the radical expression 5√26×2×35×1126×2×35×115. Next, one has to simplify the given radical expression into the simplest version or form until no further simplification shall be possible to be done.

• Step 1: First is to determine the prime factors in relation to the number under the root.
  The prime factors of 2⁶ × 2 × 3⁵ × 11 are 2, 3, 11
• Step 2: Second is to write the prime factors in the groups. 5√26×2×35×11=5√27×35×1126×2×35×115=27×35×115
• Step 3: Third is to simplify any multiplication as well as the exponents.
  5√27×35×11=5√25×22×35×11=2×35√22×1127×35×115=25×22×35×115=2×322×115
• Step 4: Fourth is simplify the radical until no more simplification shall be possible to be done.
  5√27×35×11=2×35√22×11=65√4×11=65√44

Simplifying Radical Expressions Operations

i-Adding Radicals

One should look at few examples. In this first instance, both the radicals have the same root as well as index.

This next instance contains a lot more addends. Notice the manner in which one can combine like terms (radicals that have the same root as well as index) but one cannot combine unlike terms.

Notice that the expression in the preceding instance is simplified even though it consists of 2 terms:  and . It can be said to be a mistake to try to combine them a lot more! (Few individuals make the mistake that . This is not correct because and  are not like the radicals, and hence, they cannot be added.)

Sometimes a person might be required to add as well as simplify the radical. If the specific radicals are different, then, one should try simplifying 1^st—he or she might end up being able to combine the specific radicals at the end, as demonstrated in these next 2 examples.

ii-Multiplying Radicals

1. Ensure that the radicals consist of the same index.In order to multiply the radicals through the utilization of the basic method, they should have the same index. The ‘index’ can be said to be the very small number that is written just to the left of the highest line in the specific radical symbol. If there is no existence of an index number, then, the specific radical is understood to be a specific square root (index 2) and shall be possible to be multiplied with the other square roots. One shall be able to multiply the radicals with the different indexes, but that would be a more advanced method and shall be explained later. Following can be said to be 2 examples of the multiplication through the utilization of the radicals with the same indexes: -

• Ex. 1: √(18) x √(2) = ?
• Ex. 2: √(10) x √(5) = ?
• Ex. 3: ³√(3) x ³√(9) = ?

2. Multiply the numbers under the radical signs.Next, just multiply the specific numbers under the radical or the square root signs as well as keep them there. Following is the manner in which one shall be able to do it:

• Ex. 1: √(18) x √(2) = √(36)
• Ex. 2: √(10) x √(5) = √(50)
• Ex. 3: ³√(3) x ³√(9) = ³√(27)

3. Simplify the particular radical expressions.If one has multiplied the radicals, there can be said to be a good chance that they could be simplified to the perfect squares or the perfect cubes, or that they might be simplified by finding any perfect square in the form of a factor of the final product.

iii-Rationalizing Denominators

Rationalizing the denominator actually means the particular procedure of moving any root, for example, any cube root or any square root from the bottommost part of any fraction (denominator) to the topmost part of the fraction (numerator). In this manner, it shall be possible to bring the fraction to its simplest version thereby, the denominator actually becomes rational.

How to Simplify Radicals With Exponents

A fraction could be an exponent. When any fraction is a specific exponent, one shall be able to change it so that a there would be a 1^st, 2^nd, 3^rd, etc. root of a certain thing.

For instance,

1^1/2 = square root of 1

1^1/3 = third root of 1

1^1/4 = fourth root of 1

Simplifying Radical Expressions Examples 

i-Simplify Radical fractions

Sometimes one shall be faced with any radical expression that does not have any concise answer, such as √3 from the previous instance. In that case, a person preserves radical term just as it is, utilizing basic operations such as factoring or canceling to either isolate it or remove it.

ii-Simplifying Radical square roots

Introduces radical symbol as well as concept of taking roots.

iii-Simplifying Radical with variables

Radical expressions are the expressions that contain the radicals.