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Chapter 10: Problem 16

Multiply. $$ \sqrt[5]{9 t^{2}} \sqrt[5]{2 t} $$

Short Answer

Expert verified
\( \sqrt[5]{18 t^{3}} \)

Step by step solution

01

Apply the Property of Radicals

Recall that the product of two radicals with the same index can be combined under a single radical. Here, both radicals are fifth roots. So, \( \sqrt[5]{9 t^{2}} \sqrt[5]{2 t} = \sqrt[5]{(9 t^{2})(2 t)} \).
02

Multiply the Expressions Inside the Radical

Multiply the expressions inside the radical: \( 9 t^{2} \times 2 t = 18 t^{3} \). So now we have \( \sqrt[5]{18 t^{3}} \).

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Key Concepts

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

Multiplying Radicals
When multiplying radicals, it is important to identify if they share the same index (root). If they do, you can simplify the expression by combining them under a single radical. This is often useful in problems involving roots.
In the given exercise, we multiply two fifth roots: \(\sqrt[5]{9 t^{2}} \sqrt[5]{2 t}\).
Since both have the same index (5), we can apply this key property:
\(\sqrt[5]{a} \sqrt[5]{b} = \sqrt[5]{ab}\). Here's how it works in our example:
\(\sqrt[5]{9 t^{2}} \sqrt[5]{2 t} = \sqrt[5]{(9 t^{2})(2 t)}\). This combination simplifies to one single radical expression, making it easier to solve.
Properties of Radicals
Understanding the properties of radicals simplifies computations and helps recognize patterns. Here are the key properties used in such problems:
  • Product Property of Radicals:
    For any non-negative numbers \(a\) and \(b\), and positive integer \(n\): \(\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}\). This property is crucial when dealing with the multiplication of radicals with the same index.

  • Simplifying Radicals: The process often involves combining constants and variables separately under a single root when possible.
    For example, in our solution, \(\sqrt[5]{(9 t^{2})(2 t)} = \sqrt[5]{18 t^{3}}\). To simplify it further, you would need to find if there is a way to reduce under the radical sign, though in this case, it's as simplified as it can be.

  • Radicals with Variables: Handle variables under radicals just like numbers, making sure to apply multiplication accurate to both numbers and variable parts: \(\sqrt[5]{x^m} \sqrt[5]{x^n} = \sqrt[5]{x^{m+n}}\). This is used in our problem where \(\sqrt[5]{9 t^{2}} \sqrt[5]{2 t} = \sqrt[5]{18 t^{3}}\).
Fifth Roots
The fifth root of a number or expression is the value that, when multiplied by itself five times, gives the original number or expression.
Mathematically, this is written as \(\sqrt[5]{a}\), where \(a\) is the quantity inside the radical.
In the given problem, we are dealing with fifth roots:
  • Combining Fifth Roots:
    You can combine instances of fifth roots through multiplication, provided the indices are the same. For instance, with our radicals \(\sqrt[5]{9 t^{2}} \sqrt[5]{2 t}\), we make use of the property to combine them: \(\sqrt[5]{(9 t^{2})(2 t)} = \sqrt[5]{18 t^{3}}\).

  • Simplifying Fifth Roots: As with other radicals, the goal is to see if the expression inside the root can be simplified further. With \(\sqrt[5]{18 t^{3}}\), we look for factors that can further reduce it. While 18 and \(t^{3}\) cannot be easily factored out to simplify under the fifth root, this approach remains consistent.

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Most popular questions from this chapter

Simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{-80 a^{14}} $$

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Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt[4]{x^{2} y^{3}}}{\sqrt[3]{x y}}$$

Ramon incorrectly writes $$\sqrt[5]{x^{2}} \cdot \sqrt{x^{3}}=x^{2 / 5} \cdot x^{3 / 2}=\sqrt[5]{x^{3}}$$ What mistake do you suspect he is making?

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