Master Radical Operations with Step-by-Step Solutions

A power with a fractional exponent equals a root whose index is the denominator of the fraction and the exponent of the radicand is the numerator.

To solve, we first decompose the radicand into factors, perform the operations on the radicand and extract factors

 

1

 

 

2

 

 

3

 

In this case we convert the exponent that is an exact decimal number to a fraction

 

 

4

We convert to a fraction the exponent that is a pure repeating decimal

 

 

Once we know the exponent as a fraction, we solve

 

1

 

 

The exponent of two is less than the index , therefore it remains in the radicand.

 

The exponent of equals the index , therefore the comes out of the radicand.

 

The exponent of is greater than the index , therefore this exponent is divided by the index. The quotient obtained is the exponent of the factor outside the radicand and the remainder is the exponent of the factor inside the radicand.

2

 

The exponents are greater than the index, therefore these exponents are divided by the index.

 

Each of the quotients obtained will be the exponent of the corresponding factor outside the radicand and each of the remainders obtained will be the exponent of the corresponding factor inside the radicand.

 

1

 

Before we begin solving, let's remember some properties of radicals. We know that the radical applied to a product is the product of the radicals

 

and that the index of the radical when converted to exponential form divides the power of the base

 

then these two results together we use to simplify expressions with radicals multiplied by factors, that is

 

and thus we can simply use the result

 

 

Let's now apply this process to our problem:

 

 

We introduce the raised to the index of the radical and perform the operations

 

2

 

We introduce the factors raised to the index

 

 

We perform the operations

 

We find the least common multiple of the indices, which will be the common index

We divide the common index by each of the indices and each result obtained is multiplied by its corresponding exponents

We perform the operations in the radicals

1

 

Since the radicals are like terms, we add the coefficients of the radicals:

 

 

 

2

 

We add the coefficients of the radicals:

 

 

 

3

 

We decompose the radicands into factors and extract factors from the radicals (if possible) and multiply them by the coefficient of the corresponding radical

 

 

 

4

We extract factors from the radicals and multiply them by the coefficient of the corresponding radical

 

 

 

We simplify the radicals. In the first radical we divide the index and the exponent of the radicand by , in the second by and in the third by

 

 

 

We add the coefficients of the radicals

 

 

To perform these sums of unlike radicals, we will follow these two steps:

 

We decompose the radicals into factors and extract factors from the radicals (if possible) and multiply them by the coefficient of the corresponding radical

 

 

We add the coefficients of the radicals

 

1

 

 

2

 

 

3

 

4

 

1

 

We rationalize the second term by multiplying and dividing by the square root of

 

 

We factor out the common factor of square root of and add

 

 

 

2

 

We decompose the radicals into factors

 

 

 

In the first two terms we extract factors, the third we simplify the radical by dividing the index and the exponent of the radicand by and the last we will rationalize by multiplying and dividing by the cube root of

 

 

Since all the radicals are like terms we can add their coefficients

 

 

1

 

Since the radicals have the same index we multiply the radicands and decompose into factors to extract factors from the radical.

 

 

2

 

We decompose the radicands into factors

 

 

 

We reduce to a common index so we need to calculate the least common multiple of the indices, which will be the common index.

 

 

 

We divide the common index by each of the indices and each result obtained is multiplied by its corresponding exponents .

 

We perform the product of powers with the same base in the radicand and extract factors from the radicand

 

 

 

3

 

We calculate the least common multiple of the indices.

 

 

Let's proceed with the calculations:

 

 

1

 

Since the radicals have the same index we divide the radicands and simplify the radical by dividing the index and exponent of the radicand by

 

 

 

2

 

First we reduce to a common index so we need to calculate the least common multiple of the indices, which will be the common index.

 

 

.

 

We divide the common index by each of the indices and each result obtained is multiplied by its corresponding exponents .

 

We decompose into factors to be able to perform the division of powers with the same base and divide.

 

 

3

 

We perform the same steps as in the previous exercise

 

 

 

We simplify the radical by dividing the index and the exponent of the radicand by , and finally extract factors

 

 

First we calculate the least common multiple of the indices, which will be the common index

 

 

We divide the common index by each of the indices and each result obtained is multiplied by its corresponding exponents

 

We remove the parentheses, simplify the fraction and multiply the powers with the same base in the numerator

We simplify the radical by dividing the index and the exponent of the radicand by

Finally we extract factors

First, let's note that , therefore

 

 

1

 

We raise the radicand to the second power, decompose into factors and raise them to the second power and finally extract factors

 

 

2

 

We raise the radicands to the fourth power, decompose the radicands into factors and extract from the radical

 

 

 

In the radicands we perform the operations with powers and convert to a common index to be able to perform the division

 

 

 

We simplify the radical by dividing the index and the exponents of the radicand by and perform a division of powers with the same exponent

 

 

 

We can rationalize by multiplying and dividing by the cube root of

 

 

1

 

A difference squared equals the square of the first, minus twice the first times the second, plus the square of the second

 

 

 

2

 

 

3

 

A sum times a difference equals the difference of squares

 

 

 

4

 

1

 

We perform the multiplication of fractions, in the denominator we have a sum times a difference which equals the difference of squares

 

 

 

2

 

 

The difference of squares in the denominator is written as a sum times a difference and the fraction is simplified

 

 

1

 

We multiply the indices

 

 

 

2

 

We introduce the first inside the cube root so we have to cube it and multiply the powers with the same base, we continue this procedure until all values have been introduced into the radicals, and with this, we multiply and then we get .

 

 

3

 

We introduce the inside the square root by squaring it.

We multiply the powers with the same base.

We multiply the indices and simplify by dividing the resulting index and the exponent of the radicand by 3.

 

1

We multiply the numerator and the denominator by the square root of , perform the calculations and simplify the fraction.

 

 

 

2

 

We write the radicand in power form: .

 

 

We have to multiply in the numerator and denominator by the fifth root of .

We multiply the radicals in the denominator, extract factors from the radical and simplify the fraction.

 

3

 

We multiply the numerator and the denominator by the conjugate of the denominator, remove parentheses in the numerator and perform the sum times difference in the denominator, obtaining a difference of squares.

 

 

 

In the denominator we extract the radicands and divide by , that is, we change the sign of the numerator.

 

 

 

4

 

We multiply and divide the fraction by the conjugate of the denominator.

 

 

 

We perform the sum times difference in the denominator, obtaining a difference of squares and operate:

 

 

 

5

 

We multiply the numerator and the denominator by the conjugate of the denominator, remove parentheses in the numerator and perform the sum times difference in the denominator, obtaining a difference of squares.

 

 

In the numerator we decompose into factors and extract factors, we finish by performing the operations in the denominator.

 

 

1

 

We multiply the numerator and the denominator by the square root of and perform the calculations

 

 

2

 

Here we realize that to eliminate the radical we need to generate the product

 

 

so that the radical is eliminated, that is

 

 

in other words, since we already have in the denominator, we only need to multiply it by to eliminate the radical.

So as to not affect the numerical value of the expression, we multiply in both the numerator and denominator, and this is how our development looks

 

3

 

We multiply and divide the fraction by the conjugate of the denominator.

 

 

 

In the numerator we remove parentheses and in the denominator we perform the sum times difference, obtaining a difference of squares.

 

 

 

We perform the operations and simplify the fraction by factoring out

 

 

 

4

 

We multiply and divide the fraction by the conjugate of the denominator

 

 

In the numerator we remove parentheses and in the denominator we perform the sum times difference, obtaining a difference of squares

 

 

 

5

 

We multiply the numerator and the denominator by the conjugate of the denominator

 

 

 

We write the numerator in power form

 

 

 

In the numerator we have a difference squared which equals the square of the first, minus twice the first times the second, plus the square of the second. In the denominator we have a sum times difference which equals the difference of squares

 

 

 

We perform the operations and simplify at the end