Question

\(\sqrt[4]{16 w^{10}}+2 w \sqrt[4]{w^6}\)

Short answer

The simplified form of \(\sqrt[4]{16 w^{10}}+2w \sqrt[4]{w^6}\) is \(4w^{2.5}\).

Step-by-step solution

Simplify the first radical

We start by simplifying \(\sqrt[4]{16 w^{10}}\). Consider the number and the variable separately. The fourth root of 16 is 2. The variable \(w^{10}\) can be written as \(w^{4*2.5}\) so the fourth root is \(w^{2.5}\). Thus, \(\sqrt[4]{16 w^{10}}\) simplifies to \(2w^{2.5}\).

Simplify the second radical

Next, we simplify the second term, \(2w \sqrt[4]{w^6}\). Again, consider the number and variable separately. The outside multiplier remains as it is, while \(\sqrt[4]{w^{6}}\) simplifies to \(w^{1.5}\). Thus, the term simplifies to \(2w* w^{1.5} = 2w^{2.5}\).

Combine like terms

Both pieces of the expression simplify to the same term \(2w^{2.5}\). So we add the like terms together to obtain the final result. \(2w^{2.5}+2w^{2.5}\) simplifies to \(4w^{2.5}\).

Key concepts

Fourth Root

The term 'fourth root' refers to the process of finding a number that, when multiplied by itself four times, equals a given number. In mathematical notation, the fourth root of a number is represented as \( \sqrt[4]{x} \), where x is the number in question. It's similar to the square root but instead of looking for a number that squares to the desired value, we seek the number that when raised to the fourth power gives that value.

For example, if we have \( \sqrt[4]{16} \), we are looking for a number which, when used in multiplication four times, results in 16. The answer to this is 2, as \( 2 \times 2 \times 2 \times 2 = 16 \). Likewise, when dealing with variables, we apply the same principle. If the exponent of the variable is a multiple of 4, we can take its fourth root by dividing the exponent by 4.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another number. This is denoted as \( b^n \), where \( b \) is the base and \( n \) is the exponent. The exponent signifies how many times the base is multiplied by itself.

For instance, \( w^{10} \) can be seen as \( w \) multiplied by itself 10 times. To simplify this in the context of fourth roots, we can rewrite the exponent as a product of 4 and another number, which allows us to use exponentiation rules to simplify the expression inside the radical, as seen in the exercise solution.

Like Terms

Like terms are terms within an algebraic expression that have the exact same variable factors and are raised to the same power. These terms can be combined through addition or subtraction. In our exercise, the process of combining like terms becomes simple because the simplified forms of both radical expressions result in the same term, \( 2w^{2.5} \).

When like terms are identified, you can treat them just like regular numbers in addition or subtraction. In this case, \( 2w^{2.5} + 2w^{2.5} \) combines to give \( 4w^{2.5} \), which is the sum of the coefficients (2 + 2) multiplying the common variable part.

Radical Expressions

Radical expressions contain a radical symbol (\sqrt{}), and work with roots of numbers or variables. Simplifying these expressions often involves finding perfect roots and reducing the expressions to remove the radical if possible, or making them simpler.

When simplifying radical expressions, we look to factorize the number under the radical to its prime factors or to break down the exponent of the variable to see if they can be factored by the index of the root. For example, in a fourth root expression like \( \sqrt[4]{w^{10}} \), we would look to see if 10 can be divided by 4 to simplify the expression and in this exercise, we simplified it to \( w^{2.5} \) by recognizing that 10 is 4 times 2.5.