Complex Fraction - Radicals

Category: Algebra

"Published in Suisun City, California, USA"

Simplify

Solution:

Consider the given equation above

Get the Least Common Denominator (LCD) of the numerator and the denominator of the given fraction and then simplify as follows

Get the reciprocal of the divisor and perform the multiplication, we have

Simplify the above equation and therefore, the final answer is

 

Algebraic Operations - Radicals, 27

Category: Algebra

"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. Eliminate the cube root sign at the denominator by applying the principles of Algebra which is the Sum and Difference of Two Cubes as follows

Apply the Distributive Property of Multiplication Over Addition at the numerator, we have

If you will multiply a radical with another radical with the same index, then the terms inside the radicals will be multiplied together.

How about if you will multiply a radical with another radical with different index? If their indexes are different, then you cannot multiply the terms inside the radicals together. The first two terms of the numerator at the above equation can be written as

At the first term of the numerator, the Least Common Denominator (LCD) of their fractional exponents is 12 (4 x 3). ¼ becomes 3/12 (12 ÷ 4 x 1 = 3) and ⅔ becomes 8/12 (12 ÷ 3 x 2 = 8).

At the second term of the numerator, the Least Common Denominator (LCD) of their fractional exponents is 12 (4 x 3). ¼ becomes 3/12 (12 ÷ 4 x 1 = 3) and ⅓ becomes 4/12 (12 ÷ 3 x 1 = 4).

Hence, the above equation becomes

Since the index of the two radicals at the first two terms of the numerator are now the same, then the terms inside the radicals can be multiplied together as follows

Therefore, the final answer is

 

Algebraic Operations - Radicals, 26

Category: Algebra

"Published in Newark, California, USA"

Perform the indicated operations

Solution:

Consider the given equation above

If you will multiply a radical with another radical with the same index, then the terms inside the radicals will be multiplied together.

How about if you will multiply a radical with another radical with different index? If their indexes are different, then you cannot multiply the terms inside the radicals together. The given equation above is an example of multiplication of radicals with different indexes. The given equation above can be written as

Next, convert the exponential fractions into their common denominator by getting their Least Common Denominator (LCD) as follows

The LCD of 2 and 3 is 6. ½ becomes 3/6 (6 ÷ 2 x 1 = 3) and ⅓ becomes 2/6 (6 ÷ 3 x 1 = 2). The above equation can be written as

Since the index of the two radicals are now the same, then the terms inside the radicals can be multiplied together as follows

Therefore, the final answer is