Problem 143 Write with a rational exponent. ... [FREE SOLUTION]

Chapter 8: Problem 143

Write with a rational exponent. \(\sqrt[5]{u^{2}}\)(b) \((\sqrt[3]{6 x})^{5}\)(c)\(\sqrt[4]{\left(\frac{18 a}{5 b}\right)^{7}}\)

Short Answer

Expert verified

(a) u^{2/5} (b) (6x)^{5/3} (c) (\frac{18 a}{5 b})^{7/4}

Step by step solution

Convert Radical to Exponent (Part A)

To convert \(\sqrt[5]{u^{2}}\) into a rational exponent, use the rule \(\sqrt[n]{a^{m}} = a^{m/n}\). Therefore, \(\sqrt[5]{u^{2}} = u^{2/5}\).

Convert Radical to Exponent (Part B)

For \((\sqrt[3]{6 x})^{5}\), convert the inner radical first using the same rule \(\sqrt[n]{a} = a^{1/n}\). So, \(\sqrt[3]{6 x} = (6x)^{1/3}\). Then raise this to the power of 5: \(((6x)^{1/3})^{5} = (6x)^{5/3}\).

Convert Radical to Exponent (Part C)

For \(\sqrt[4]{\left(\frac{18 a}{5 b}\right)^{7}}\), convert the inner radical using the rule \(\sqrt[n]{a^{m}} = a^{m/n}\). So, \(\sqrt[4]{\left(\frac{18 a}{5 b}\right)^{7}} = (\frac{18 a}{5 b})^{7/4}\).

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Key Concepts

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

radicals

Radicals represent roots of numbers or expressions. When you see \(\root[n]{a}\), it means you are looking for a number that, when raised to the power n, gives a. For example, \(\root[3]{27}\) is 3, because 3 * 3 * 3 = 27. Common radicals include square roots \(\root[2]{a}\) and cube roots \(\root[3]{a}\). Understanding radicals helps in simplifying and solving many algebra problems.
When converting radicals to exponents, use the rule: \(\root[n]{a^m} = a^{m/n}\). This will help in easier manipulation of algebraic expressions. For example, \(\root[4]{a^7} = a^{7/4}\).

exponent rules

Exponent rules are essential for simplifying expressions involving powers. Here are some key rules:

Product of powers: \(a^m \times a^n = a^{m+n}\)
Quotient of powers: \(a^m / a^n = a^{m-n}\) for \(a eq 0\)
Power of a power: \((a^m)^n = a^{mn}\)
Power of a product: \((ab)^m = a^m \times b^m\)
Power of a fraction: \(\root[n]{\frac{a}{b}} = \frac{\root[n]{a}}{\root[n]{b}}\)

For example, in the exercise above, converting \((\root[3]{6x})^5\) to \((6x)^{5/3}\) uses both the power of a product and the power of a power rules.

simplifying expressions

Simplifying expressions makes them easier to work with. It involves combining like terms, reducing fractions, and applying exponent rules. For instance, in the given exercise:

Part (a): \(\root[5]{u^2}\) becomes \(u^{2/5}\)
Part (b): \((\root[3]{6x})^5\) simplifies to \((6x)^{5/3}\)
Part (c): \(\root[4]{(\frac{18a}{5b})^7}\) converts to \((\frac{18a}{5b})^{7/4}\)

Each of these steps uses rational exponents to make the expressions simpler and more manageable. By understanding and applying these rules, you'll be well-equipped to handle complex algebraic problems.

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91影视

Write with a rational exponent. \(\sqrt[5]{u^{2}}\)(b) \((\sqrt[3]{6 x})^{5}\)(c)\(\sqrt[4]{\left(\frac{18 a}{5 b}\right)^{7}}\)

Short Answer

Step by step solution

Convert Radical to Exponent (Part A)

Convert Radical to Exponent (Part B)

Convert Radical to Exponent (Part C)

Key Concepts

radicals

exponent rules

simplifying expressions

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