Problem 14
Question
Write each expression in radical form. Assume that all variables represent positive real numbers. $$-\sqrt[4]{z^{5}}$$
Step-by-Step Solution
Verified Answer
The expression in radical form is
\(-z \sqrt[4]{z}\).
1Step 1: Understand the Expression
The given expression is \(-\sqrt[4]{z^{5}}\). We need to rewrite this expression in radical form, which involves expressing powers and roots using radicals.
2Step 2: Break Down the Expression
The expression \(-\sqrt[4]{z^{5}}\) is an expression with a fourth root. The number inside the root, \(z^5\), has a power greater than the root degree, which we'll need to manage in the radical form. We do this by separating \(z^5\) into \(z^4 \times z^1\).
3Step 3: Apply the Radical to Each Component
Rewrite \(-\sqrt[4]{z^5}\) using the separated components: \(-\sqrt[4]{z^4 \times z^1}\). By properties of radicals, this can be split into two parts: \(-\sqrt[4]{z^4} \cdot \sqrt[4]{z}\).
4Step 4: Simplify the Radicals
Simplify \(\sqrt[4]{z^4}\) to \(z\) since the fourth root and the fourth power cancel each other out. Then, we have \(-z \cdot \sqrt[4]{z}\).
5Step 5: Write the Final Radical Form
Combine the simplified parts to write the expression as \(-z \sqrt[4]{z}\). This is the expression written in radical form, maintaining the negative sign from the original expression.
Key Concepts
ExpressionsRoots and PowersSimplifying Radicals
Expressions
An expression in mathematics is a combination of symbols that is used to represent a value. In our exercise, the expression is \(-\sqrt[4]{z^{5}}\). This particular expression involves a root, which is a common component in algebra.
Expressions can be transformed or simplified using various rules from algebra. For instance, you can add, subtract, multiply, or divide different parts of an expression. With this example, the goal is to express it in a different form - the radical form.
The key is to recognize different components, such as roots and powers, and understand how they relate to each other. Observing the relationship between roots and powers allows us to manipulate the expression into the radical format required.
Expressions can be transformed or simplified using various rules from algebra. For instance, you can add, subtract, multiply, or divide different parts of an expression. With this example, the goal is to express it in a different form - the radical form.
The key is to recognize different components, such as roots and powers, and understand how they relate to each other. Observing the relationship between roots and powers allows us to manipulate the expression into the radical format required.
Roots and Powers
Roots and powers are fundamental concepts in mathematics that often go hand-in-hand. A power refers to the number that results when a base is multiplied by itself a certain number of times. For example, in \(z^5\), \(z\) is multiplied by itself five times.
On the other hand, a root is the opposite operation of a power. It involves finding a number which, when raised to a specified power, will give the original number inside the root. In our problem, the fourth root is sought because we are dealing with \(-\sqrt[4]{z^{5}}\).
Understanding the operation of roots and powers is crucial for simplifying expressions, as we did when we broke down the expression into \(z^{4}\) and \(z^{1}\). From there, \(\sqrt[4]{z^4}\) converts neatly to \(z\) because \(z^4\) raised to the fourth root returns to its base \(z\).
On the other hand, a root is the opposite operation of a power. It involves finding a number which, when raised to a specified power, will give the original number inside the root. In our problem, the fourth root is sought because we are dealing with \(-\sqrt[4]{z^{5}}\).
Understanding the operation of roots and powers is crucial for simplifying expressions, as we did when we broke down the expression into \(z^{4}\) and \(z^{1}\). From there, \(\sqrt[4]{z^4}\) converts neatly to \(z\) because \(z^4\) raised to the fourth root returns to its base \(z\).
Simplifying Radicals
Simplifying radicals involves making an expression under a root simpler, and is a frequent task in algebra.
For the given expression \(-\sqrt[4]{z^{5}}\), we simplified it by first breaking down \(z^5\) into components: \(z^4\) and \(z\). Using properties of radicals, these were expressed separately as \(\sqrt[4]{z^4} \cdot \sqrt[4]{z}\).
The next step was to simplify \(\sqrt[4]{z^4}\) to \(z\), leveraging the fact that raising something to a root, and then taking the same power, returns the original base. This step avoids any complex calculations and uses theoretical understanding.
By structuring the radicals this way, the expression transformed to \(-z \sqrt[4]{z}\). Simplifying radicals is about recognizing these opportunities to "cancel out" operations, making it easier to work with expressions.
For the given expression \(-\sqrt[4]{z^{5}}\), we simplified it by first breaking down \(z^5\) into components: \(z^4\) and \(z\). Using properties of radicals, these were expressed separately as \(\sqrt[4]{z^4} \cdot \sqrt[4]{z}\).
The next step was to simplify \(\sqrt[4]{z^4}\) to \(z\), leveraging the fact that raising something to a root, and then taking the same power, returns the original base. This step avoids any complex calculations and uses theoretical understanding.
By structuring the radicals this way, the expression transformed to \(-z \sqrt[4]{z}\). Simplifying radicals is about recognizing these opportunities to "cancel out" operations, making it easier to work with expressions.
Other exercises in this chapter
Problem 13
Write each rational expression in lowest terms. $$\frac{3(t+5)}{(t+5)(t-3)}$$
View solution Problem 13
Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none o
View solution Problem 14
Simplify each expression. Assume that all variables represent positive real numbers. $$25^{1 / 2}$$
View solution Problem 14
Factor each polynomial by grouping. $$10 a b-6 b+35 a-21$$
View solution