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Chapter 8: Problem 74

Simplify. $$\left(y^{\frac{1}{6}}\right)^{\frac{1}{3}}$$

Short Answer

Expert verified
Simplified expression: \(y^{\frac{1}{18}}\).

Step by step solution

01

Identify the Exponent Rule

Recognize that the expression \(\big(y^{\frac{1}{6}}\big)^{\frac{1}{3}}\) involves an exponentiation of an exponent. Use the power rule of exponents which states \(\big(a^m\big)^n = a^{m \times n}\).
02

Apply the Power Rule

Apply the power rule to the expression by multiplying the exponents: \(y^{\frac{1}{6} \times \frac{1}{3}}\).
03

Simplify the Exponents

Multiply the exponents \(\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}\). This simplifies the expression to \(y^{\frac{1}{18}}\).

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Key Concepts

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

Power Rule
The Power Rule is a fundamental concept when working with exponents. This rule states that when you take an exponent and raise it to another exponent, you multiply the exponents. Mathematically, it is expressed as \(\left(a^m\right)^n = a^{m \cdot n}\).

For instance, if you have an expression \(\left(y^{\frac{1}{6}}\right)^{\frac{1}{3}}\), you can simplify it using the power rule. You multiply the exponents \(\frac{1}{6} \cdot \frac{1}{3}\), resulting in a new exponent of \(\frac{1}{18}\). This simplifies the expression to \(y^{\frac{1}{18}}\).

Remember to always multiply the exponents when applying the power rule.
  • Identify the base and the exponents
  • Multiply the exponents together
  • Simplify the expression
Exponentiation
Exponentiation is the operation of raising a number (the base) to a power (the exponent). It involves repeated multiplication of the base. For example, \(2^3\) means multiplying 2 by itself three times, resulting in 8.

When it comes to powers of fractions or rational numbers, the same rules apply. In our example, \(\left(y^{\frac{1}{6}}\right)^{\frac{1}{3}}\), exponentiation is used to raise \(y^{\frac{1}{6}}\) to the power of \(\frac{1}{3}\).

To handle such expressions, you might have to:
  • Identify the base and its initial exponent
  • Raise the expression to the new exponent
  • Use the power rule to simplify the result

By understanding exponentiation, you can handle various complex expressions and simplify them efficiently.
Simplifying Exponents
Simplifying exponents is a crucial skill in algebra and higher mathematics. It allows you to reduce expressions to their simplest form for easier manipulation and understanding.

In the given problem, we start with \(\left(y^{\frac{1}{6}}\right)^{\frac{1}{3}}\). We apply the power rule by multiplying the exponents: \(\frac{1}{6} \cdot \frac{1}{3} = \frac{1}{18}\). This simplifies the expression to \(y^{\frac{1}{18}}\).

Here are steps to follow for simplifying exponents:
  • Apply exponent rules: power rule, product rule, and quotient rule
  • Combine or reduce exponents where possible
  • Always express your final answer in the simplest form

Simplifying exponents can make challenging problems more straightforward and solvable, making it an essential tool in your math toolkit.

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Most popular questions from this chapter

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Evaluate. $$\sqrt[3]{-8}$$
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