Problem 75 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. $$ 4 \sqrt[5]{2}+3 \sqrt[5]{2} $$

Step-by-Step Solution

Answer

\(7 \sqrt[5]{2}\)

Step 1 : Identify Similar Terms

Review the expression and identify that both terms \(4 \sqrt[5]{2}\) and \(3 \sqrt[5]{2}\) have similar structure, that is same base of 2 and same root of 5.

Step 2 : Add Similar Terms

Now add these similar terms together just like adding \(4x\) and \(3x\) together to get \(7x\). So, \(4 \sqrt[5]{2} + 3 \sqrt[5]{2} = 7 \sqrt[5]{2}\).

Key Concepts

Adding Radicals

Adding radicals is similar to adding like terms in algebra, such as when you combine coefficients of variables. To add radicals, you must ensure that the radicals you are adding are "like radicals." Like radicals have the same radicand (the number under the radical sign) and the same index (the type of root, like square root, cube root, etc.).
For example, in the expression \(4 \sqrt[5]{2} + 3 \sqrt[5]{2}\), both terms have a radicand of 2 and an index of 5, making them like radicals.

• Verify both the radicand and index match.
• When they do, simply add the coefficients (the numerical part that multiplies the radical).
• The result maintains the same radical form.

In our case, add the coefficients 4 and 3, resulting in \(7 \sqrt[5]{2}\). This approach is applicable in scenarios dealing with any type of roots, provided the radicals have matching radicands and indices.

Fifth Roots

The fifth root of a number is a value that, when multiplied by itself four more times (five multiplies in total), equals the original number. The notation for the fifth root of a number is \(\sqrt[5]{x}\).
To clarify, if \(b^5 = a\), then \(b = \sqrt[5]{a}\). For example, the fifth root of 32 is 2 because \(2^5 = 32\). When working with fifth roots:

• Identify any like fifth roots. Like fifth roots have the same radicand.
• Recognize that operations such as addition and subtraction can only take place with like terms.
• Remain consistent with radical manipulation rules.

Grasping the concept of the fifth root requires practice, especially when simplifying complex expressions involving radicals.

Like Terms

"Like terms" is a fundamental concept in algebra that refers to terms that have the same variable raised to the same power, which allows them to be combined. This principle extends to radicals as well, where the radical part must be identical for terms to be considered like terms.
In the expression \(4 \sqrt[5]{2} + 3 \sqrt[5]{2}\), both terms are like terms because they share the same radical, \(\sqrt[5]{2}\).

• Like terms allow for straightforward addition or subtraction of coefficients.
• Checking for like terms can simplify expressions significantly.
• The resulting expression after combining like terms maintains the same radical form.

Approaching problems with an eye for like terms can streamline solving techniques and make complex expressions more manageable.