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Chapter 8: Problem 25

Simplify. $$8 \sqrt[4]{z}+3 \sqrt[4]{z}$$

Short Answer

Expert verified
11 \sqrt[4]{z}

Step by step solution

01

Identify Like Terms

Notice that both terms, \(8 \sqrt[4]{z} \) and \(3 \sqrt[4]{z} \), contain the same fourth root expression, \( \sqrt[4]{z} \). Hence, they are like terms and can be combined.
02

Combine the Coefficients

Add the coefficients of the like terms together. This means adding 8 and 3. \(8 + 3 = 11\).
03

Rewrite the Expression

Combine the coefficients with the common fourth root expression. Thus, the expression simplifies to \(11 \sqrt[4]{z} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Like Terms
Like terms are terms that have the same variable raised to the same power. When you encounter expressions with like terms, it means you can simplify them by combining their coefficients.
In the exercise, both terms are \(8 \sqrt[4]{z} \) and \(3 \sqrt[4]{z}\). These terms are like terms because they both involve the fourth root of \( z \).
Similarly, if you had \(5x \) and \(2x \), they are like terms because they both have the variable \( x \). This property of like terms allows us to simplify the expression by merging the coefficients.
Exploring the Fourth Root
The fourth root of a number is a value that, when multiplied by itself four times, equals the original number. Mathematically, the fourth root of a number \(a\) is represented as \(\sqrt[4]{a}\).
For instance, \(\sqrt[4]{16} = 2\) because \(2^4 = 16\).
Recognizing the fourth root can be helpful when working with radical expressions. In our exercise, \(\sqrt[4]{z}\) signifies finding the value which, when raised to the power four, equals \(z\). Understanding this concept is crucial to simplify or combine expressions that involve fourth roots.
Combining Coefficients
To simplify expressions with like terms, we combine their coefficients. The coefficient is the numerical factor multiplying the variable or radical expression.
In the given problem, we have two terms \(8 \sqrt[4]{z}\) and \(3 \sqrt[4]{z}\). Here, 8 and 3 are the coefficients.
When combining these like terms, you add their coefficients together: \(8 + 3 = 11\).
This results in the simplified expression \(11 \sqrt[4]{z}\). By understanding how to find and combine coefficients, you can simplify many algebraic expressions easily.

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