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Chapter 10: Problem 136

\(\sqrt[3]{m^{9}}\)

Short Answer

Expert verified
The simplified form of \(\sqrt[3]{m^{9}}\) is \(m^3\).

Step by step solution

01

Identify the Given Expression

The expression provided is \(\sqrt[3]{m^{9}}\). This is a cube root operation.
02

Rewrite the Radicand with Exponents

Express the radicand \(m^9\) in a more manageable form. Recall that \(m^9 = (m^3)^3\). Hence, \(\sqrt[3]{m^{9}} = \sqrt[3]{(m^3)^3}\).
03

Simplify the Cube Root

Use the property of cube roots \(\sqrt[3]{a^3} = a\) to simplify the expression. Therefore, \(\sqrt[3]{(m^3)^3} = m^3\).

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Key Concepts

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

Understanding Cube Roots
A cube root refers to the number which, when multiplied by itself three times, gives the original number. For instance, the cube root of 8 is 2, because 2 multiplied by itself three times equals 8: 
2 \times 2 \times 2 = 8
If we have a number or variable inside the cube root, like \(\sqrt[3]{m^9}\), it means we're looking for what number raised to the power of three will give us \(m^9\).
In this case, we can simplify this by breaking down the exponentiation process.
Exploring Exponents
Exponents describe how many times a number or variable is multiplied by itself. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\). In the given problem, we're dealing with \(m^9\) which means the variable m is multiplied by itself nine times: \(m \times m \times m \times m \times m \times m \times m \times m \times m\).
To simplify finding the cube root, we notice that 9 can be written as 3 multiplied by 3, hence \(m^9 = (m^3)^3\). This knowledge will be beneficial in simplifying the problem by breaking it into smaller, manageable parts.
What is a Radicand?
The radicand is the number or expression inside a root symbol. In our problem, the radicand is \(m^9\) under the cube root symbol.
This makes understanding and simplifying the radicand crucial since it directly impacts the outcome when the cube root operation is applied.

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