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Chapter 7: Problem 15

Multiply. $$\sqrt[5]{8 y^{3}} \sqrt[5]{10 y}$$

Short Answer

Expert verified
\(\sqrt[5]{80y^4}\)

Step by step solution

01

- Express in Radical Form

Rewrite the given expression in radical form. The expression is given as \(\sqrt[5]{8 y^{3}} \sqrt[5]{10 y}\). Keep the radical form for now.
02

- Combine Radicals

Using the property of radicals that \(\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{a \cdot b}\), combine the radicals: \(\sqrt[5]{8y^3} \sqrt[5]{10y}\) becomes \(\sqrt[5]{(8y^3)\cdot(10y)}\).
03

- Multiply Inside the Radical

Multiply the expressions inside the radical: \((8y^3) \cdot (10y)\) equals \((8 \cdot 10) \cdot (y^3 \cdot y) = 80y^4\). So now, the expression becomes \(\sqrt[5]{80y^4}\).
04

- Simplify the Radical Expression

Leave the expression as a simplified radical unless further instructions indicate to solve more: the simplified form of the given expression is \(\sqrt[5]{80y^4}\).

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Key Concepts

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

Combining Radicals
In order to multiply radicals with the same index, use the property that states: \(\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{a \cdot b}\).
This allows us to combine separate radicals into a single radical expression.
For instance, in our problem, we have \(\sqrt[5]{8y^3} \sqrt[5]{10y}\).
Applying the property of combining radicals, we get: \(\sqrt[5]{8y^3 \cdot 10y}\).
This property is very useful and makes it easier to manage and simplify the expression later.
Multiplication Inside Radicals
When you multiply expressions inside a radical, it鈥檚 important to multiply both the numerical and the variable parts separately.
Let's break it down using our problem: \( (8y^3) \cdot (10y) \).
First, multiply the numerical coefficients: \(8 \cdot 10 = 80\).
Next, for the variables, use the property \(x^m \cdot x^n = x^{m+n}\).
So, \( y^3 \cdot y = y^{3+1} = y^4\).
Combining both results, we now have: \( \sqrt[5]{80y^4} \).
This simplification keeps your expression neater and more manageable.
Simplification of Radicals
The last step in our problem is to simplify the radical, as needed.
We already have the expression \( \sqrt[5]{80y^4} \) from our previous step.
For this problem, we can leave the expression as it is unless further instructions specify otherwise.
Understanding when and how to simplify radicals can make your work much clearer.
It鈥檚 a useful skill to ensure your answers are as simplified as possible while still correct.
This helps in avoiding unnecessary complications and is also a good practice for clear mathematical communication.

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