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Chapter 10: Problem 46

Simplify. See Examples 3 and 4 $$ \sqrt[3]{y^{5}} $$

Short Answer

Expert verified
\( \sqrt[3]{y^5} = y^{5/3} \).

Step by step solution

01

Identify the Expression

The given expression is \( \sqrt[3]{y^{5}} \). This is a cube root involving a power of \( y \).
02

Rewrite the Expression Using Fractional Exponents

Recall that a cube root of a number or variable \( x \) can be written as \( x^{1/3} \). Therefore, \( \sqrt[3]{y^{5}} \) can be rewritten as \( (y^5)^{1/3} \).
03

Apply the Exponent Multiplication Rule

When you have a power of a power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\). Apply this to the expression: \( (y^5)^{1/3} = y^{5 \cdot 1/3} \).
04

Simplify the Exponents

Calculate the exponent: \( 5 \cdot \frac{1}{3} = \frac{5}{3} \). Thus, the expression simplifies to \( y^{5/3} \).

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Key Concepts

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

Cube Root
A cube root is a special type of root that undoes the operation of cubing a number. When you take the cube root of a number, you're looking for a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because
  • 3 multiplied by 3 is 9, and
  • 9 multiplied by 3 is 27.
Cube roots are represented using the radical symbol with an index of 3, like this: \[ \sqrt[3]{x} \]When dealing with variables raised to powers under a cube root, like \( \sqrt[3]{y^5} \), rewriting them using fractional exponents often simplifies the process.
Fractional Exponents
Fractional exponents provide an alternative way to express roots, which can make complex expressions simpler to handle. The general idea is to express the root as a fraction in the exponent. For example,
  • a square root can be written as \( x^{1/2} \), and
  • a cube root can be written as \( x^{1/3} \).
So, whenever you see a cube root, you can think "exponent of one-third." Rewriting \( \sqrt[3]{y^5} \) using fractional exponents turns it into \( (y^5)^{1/3} \). This transformation makes it easier to apply exponent rules in following steps. Remember, fractional exponents are just another way to express roots, where the denominator of the fraction corresponds to the degree of the root.
Exponent Multiplication Rule
The exponent multiplication rule is key when simplifying expressions involving powers of powers. You use this rule when you have something like \( (a^m)^n \). The rule states that you multiply the exponents: \( a^{m \cdot n} \).Let's apply this rule to our example, \( (y^5)^{1/3} \).
  • The base \( y \) stays the same.
  • The exponents 5 and \( 1/3 \) are multiplied together.
  • This gives us \( y^{5 \cdot (1/3)} = y^{5/3} \).
This simplification shows how using exponentiation rules can transform an expression with a root into a more straightforward variable power expression. Knowing when to use these rules is essential for simplifying expressions efficiently.

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Most popular questions from this chapter

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{y^{11 / 3}}{\left(y^{5}\right)^{1 / 3}} $$

Fill in each box with the correct expression $$ \square \cdot x^{1 / 8}=x^{4 / 8}, \text { or } x^{1 / 2} $$

Which of the following are not real numbers? $$\sqrt[3]{-17}$$

Use rational expressions to write as a single radical expression. $$ \sqrt[6]{y} \cdot \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} $$

Which of the following are not real numbers? $$\sqrt[15]{-17}$$
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