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Chapter 8: Problem 82

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{y^{3}} $$

Short Answer

Expert verified
The simplified form is \( y \sqrt{y}\).

Step by step solution

01

- Understand the Radical Expression

We need to simplify the given radical expression \(\sqrt{y^3}\). Notice that \(\sqrt{y^3}\) is a square root of \(\ y^3\).
02

- Factor the Expression Inside the Radical

Rewrite \(\ y^3\) inside the square root as \(\ y^2 \times y\). This gives us: \( \sqrt{y^3} = \sqrt{y^2 \times y}\).
03

- Use the Product Property of Square Roots

By the product property of square roots \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can separate this into two square roots: \(\sqrt{y^2} \times \sqrt{y}\).
04

- Simplify Each Part

We know that \(\sqrt{y^2}\) simplifies to \( y \) because the square root and the square cancel each other out. Thus, \( \sqrt{y^2} \times \sqrt{y}\) becomes \( y \times \sqrt{y}\).
05

- Write the Final Answer

Combine the simplified parts to give the final simplified radical expression: \( y \sqrt{y}\).

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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 or cube roots. A square root undoes squaring a number, while a cube root undoes cubing a number. The symbol for square root is \( \sqrt{ } \), and for cube root, it's \( \sqrt[3]{ } \). Radicals can also have variables under the root; for example, \( \sqrt{y^3} \) is a radical expression.
Product Property of Square Roots
The product property of square roots helps in breaking down complex radicals. It states \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This means if you have the square root of a product, you can split it into the product of the square root of each factor. For example:
\( \sqrt{y^3} = \sqrt{y^2 \cdot y} \).
Then it can be separated into two roots: \( \sqrt{y^2} \cdot \sqrt{y} \).
Factoring Expressions
Factoring is the process of breaking down an expression into simpler parts. In our exercise, we factored \( y^3 \) into \( y^2 \cdot y \). This helped us use the product property of square roots. To factor means to write a number or expression as a product. For example, \( y^3 = y^2 \cdot y \), and \( 15 = 3 \cdot 5 \). Factoring simplifies working with radicals and other mathematical operations.

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Most popular questions from this chapter

Solve each equation. $$ x^{2}+4 x+3=0 $$

Simplify each radical expression. $$ \frac{30 \sqrt{10}}{5 \sqrt{2}} $$

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{a^{13}} $$

To estimate the speed at which a car was traveling at the time of an accident, a police officer drives the car under conditions similar to those during which the accident took place and then skids to a stop. If the car is driven at 30 mph, then the speed \(s\) at the time of the accident is given by $$ s=30 \sqrt{\frac{a}{p}} $$ where \(a\) is the length of the skid marks left at the time of the accident and \(p\) is the length of the skid marks in the police test. Find \(s\) for the following values of \(a\) and \(p .\) (a) \(a=862 \mathrm{ft} ; p=156 \mathrm{ft}\) (b) \(a=382 \mathrm{ft} ; p=96 \mathrm{ft}\) (c) \(a=84 \mathrm{ft} ; p=26 \mathrm{ft}\)

Perform each operation and express the answer in simplest form. $$ 2 \sqrt[4]{2}(3 \sqrt[4]{8}+5 \sqrt[4]{4}) $$
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