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Chapter 10: Problem 46

Write an equivalent expression using exponential notation. $$ (\sqrt[6]{2 a^{5} b})^{7} $$

Short Answer

Expert verified
2^{\frac{7}{6}} a^{\frac{35}{6}} b^{\frac{7}{6}}

Step by step solution

01

Rewrite in Radical Form

Rewrite the expression \(\sqrt[6]{2 a^{5} b}\) using exponential notation. Remember that \(\sqrt[n]{x}\) is the same as \(x^{\frac{1}{n}}\). Therefore, \(\sqrt[6]{2 a^{5} b} = \left(2 a^{5} b\right)^{\frac{1}{6}}\).
02

Apply Exponent

Apply the outer exponent by raising the entire expression to the 7th power. This gives: \[\left((2 a^{5} b)^{\frac{1}{6}}\right)^{7}\].
03

Multiply the Exponents

When raising a power to another power, multiply the exponents. This results in: \[\left(2 a^{5} b\right)^{\frac{7}{6}}\].
04

Distribute the Exponent

Distribute the \frac{7}{6}\ exponent to each factor inside the parentheses: \[\left(2\right)^{\frac{7}{6}} \left(a^{5}\right)^{\frac{7}{6}} \left(b\right)^{\frac{7}{6}}\].
05

Simplify Each Term

Simplify each term individually. This results in: \[2^{\frac{7}{6}} a^{\frac{35}{6}} b^{\frac{7}{6}}\].

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Key Concepts

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

algebra
Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's a powerful tool that allows us to generalize arithmetic by using letters to represent numbers. This generalization enables us to solve equations and understand relationships between different quantities. In this exercise, algebra is used to simplify and transform the given expression step by step.
radical expressions
Radical expressions involve roots, such as square roots, cube roots, etc. A radical can often be rewritten using exponents. For example, \(\backslash\text{\sqrt[n]{x}}\) is the same as \(x^\{\frac{1}{n}}\). This is helpful when we need to perform operations that are easier when working with exponents.
In the given problem: The expression \(\sqrt[6]{2 a^{5} b}\) can be rewritten as \(\left(2 a^{5} b\right)^{\frac{1}{6}}\), transforming the radical into an exponential form which simplifies further operations.
exponent rules
Exponent rules help us manipulate and simplify expressions that involve powers of numbers or expressions.
Here are some key rules:
  • Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power Rule: \(\left(a^m\right)^n = a^{m\times n}\)
  • Power of a Product Rule: \[\left(ab\right)^m = a^m b^m\]

In our exercise, when raising a power to another power, we apply the Power of a Power Rule. This means multiplying exponents: \[\left((2 a^{5} b)^{\frac{1}{6}}\right)^{7} = \left(2 a^{5} b\right)^{\frac{7}{6}}\]. Next, we distribute the \(\frac{7}{6}\) exponent to each factor inside the parentheses:
\[\left(2\right)^{\frac{7}{6}} \left(a^{5}\right)^{\frac{7}{6}} \left(b\right)^{\frac{7}{6}}\]
And finally simplify each term individually to get:
\[2^{\frac{7}{6}} a^{\frac{35}{6}} b^{\frac{7}{6}}\]

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Most popular questions from this chapter

Review simplifying rational expressions (Sections\(7.1-7.5)\) Perform the indicated operation and, if possible, simplify. $$ \frac{15 a^{2} x}{8 b} \cdot \frac{24 b^{2} x}{5 a}[7.2] $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\sqrt[3]{x^{2} y}(\sqrt{x y}-\sqrt[5]{x y^{3}})$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt[3]{x^{2}}}{\sqrt[5]{x}}$$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ 3 \sqrt{2 x^{5}} \cdot 4 \sqrt{10 x^{2}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$(r-\sqrt[4]{s^{3}})(3 r-\sqrt[5]{s})$$
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