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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
