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Chapter 10: Problem 41

Write an equivalent expression using exponential notation. $$ \sqrt[5]{x y^{2} z} $$

Short Answer

Expert verified
\(x^{1/5} y^{2/5} z^{1/5}\)

Step by step solution

01

- Understanding the Radical Notation

The given expression is in radical form: \(\sqrt[5]{x y^{2} z}\). It's a fifth root of \(x y^{2} z\).
02

- Converting to Exponential Form

A radical expression \(\sqrt[5]{a}\) can be rewritten as \(a^{1/5}\) using exponential notation.
03

- Applying Exponential Notation to Each Factor

Apply the rule to each factor in the expression: \(x^{1/5}\), \((y^{2})^{1/5}\), and \(z^{1/5}\).
04

- Simplifying the Expression

Simplify each term individually: \(x^{1/5}\) remains the same, \((y^{2})^{1/5}\) can be simplified using the power rule \((a^m)^n = a^{m*n}\), resulting in \(y^{2/5}\), and \(z^{1/5}\) remains the same.
05

- Forming the Final Expression

Combine all simplified terms: \(x^{1/5} y^{2/5} z^{1/5}\).

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Key Concepts

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

Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, and so on. When you see a root symbol (鈭�), this is known as a radical.
For example, the given expression \(\sqrt[5]{x y^{2} z}\) is a radical expression with a fifth root. Understanding how to deal with these expressions is key to solving them. To convert a radical expression to its exponential form, use the fact that the n-th root of a number is the same as raising that number to the power of 1/n.
This means \(\sqrt[5]{a}\) can be rewritten as \(a^{1/5}\). Applying this to our expression, each term under the fifth root will be converted to an exponent of 1/5.
Power Rule
The power rule in exponents is an important concept to simplify expressions involving multiple exponents. The power rule states that \((a^m)^n = a^{m*n}br>\) In the context of our problem, we need this rule to simplify \((y^{2})^{1/5}\). By applying the power rule, we multiply the exponents: \((y^{2})^{1/5} = y^{(2 \cdot 1/5)}\), which simplifies to \(y^{2/5}\). This helps in rewriting radical expressions into a more manageable form.
Simplifying Expressions
Simplifying expressions involves breaking down complex algebraic terms into simpler forms. In our example, once we have each factor in its exponential notation, we can combine them to get a final expression.
Our original expression was \(\sqrt[5]{x y^{2} z}\). After using the conversion method and the power rule, we rewrite it as \(x^{1/5}, y^{2/5}, z^{1/5}\).
Finally, we combine these factors to form the simplified expression: \(x^{1/5} y^{2/5} z^{1/5}\).

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Most popular questions from this chapter

Add or subtract. Simplify by combining like radical terms, if possible. Assume that all variables and radicands represent positive real numbers. $$14 \sqrt[5]{2}-6 \sqrt[5]{2}$$

Perform the indicated operation and simplify. Write each answer in the form \(a+b i\) $$ (-5-i)-(7+4 i) $$

Complete sentence with the best choice from the column on the right. In any ____ triangle, the square of the length of the hypotenuse is the sum of the squares of the lengths of the legs. a) Hypotenuse b) Isosceles c) Legs d) Right e) Square roots f) \(30^{\circ}-60^{\circ}-90^{\circ}\)

Use rational exponents to simplify. Do not use fraction exponents in the final answer. $$ (\sqrt[7]{x y})^{14} $$

Identify the radicand and the index for each expression. $$ \frac{a}{b} \sqrt[6]{a^{2}+1} $$
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