Algebraic Operations - Radicals, 17

Category: Algebra, Arithmetic

"Published in Newark, California, USA"

Perform the indicated operations

Solution:

Consider the given equation above

The above equation can be written as

As a rule in Mathematics, all radicals in the denominator should be rationalized or eliminated. In this case, multiply both the numerator and the denominator by √3 so that √3 becomes 3 in the denominator as follows 

Therefore, the final answer is

Algebraic Operations - Radicals, 16

Category: Algebra

"Published in Newark, California, USA"

Perform the indicated operations

Solution:

Consider the given equation above

If you will cube a radical in which the index is 3 or that have a cube root sign, then the cube root sign in the equation will be cancelled.  

If you will multiply a radical with another radical with the same index, then the terms inside the radicals will be multiplied together. In this case, the given above equation can be written as follows

Apply the principles of Binomial Theorem or cubing of a binomial to the above equation as follows

 

Therefore, the final answer is

 

Algebraic Operations - Radicals, 15

Category: Algebra

"Published in Newark, California, USA"

Perform the indicated operations

Solution:

Consider the given equation above

If you will cube a radical in which the index is 2 or that have a square root sign, then the terms inside the square root sign will be raised to a third power. 

If you will multiply a radical with another radical with the same index, then the terms inside the radicals will be multiplied together. In this case, the given above equation can be written as follows 

Apply the principles of Binomial Theorem or squaring of a binomial to the above equation as follows

At the first term, x³ is not a perfect square, the factors of x³ are x² and x. x² is a perfect square. 

At the second term, x² is a perfect square. The square root of x² is x. 

At the third term, 4y² is a perfect square. The square root of 4y² is 2y. 

At the fourth term, 8y³ is not a perfect square, the factors of 8y³ are 4y² and 2y. 4y² is a perfect square. 

Hence, the given equation above becomes

Therefore, the final answer is