Problem 451 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{a^{3}}\) (b) \(\sqrt[12]{b^{12}}\)

Step-by-Step Solution

Answer

a, b

Step 1: Identify the Radical Expression for Part (a)

Given the expression \(\root{3}{a^3}\). Identify that the expression is a cube root of \(a^3\).

Step 2: Apply the Property of Radicals

Using the property \(\root{n}{a^n} = a\) when n is the index of the root, simplify \(\root{3}{a^3}\) directly to \(a\).

Step 3: Identify the Radical Expression for Part (b)

Given the expression \(\root{12}{b^{12}}\). Identify that the expression is a 12th root of \(b^{12}\).

Step 4: Apply the Property of Radicals Again

Using the property \(\root{n}{a^n} = a\) again, simplify \(\root{12}{b^{12}}\) directly to \(b\).

Step 5: Write the Simplified Results

Combine the results of each part: The simplified form of \(\root{3}{a^3}\) is \(a\), and the simplified form of \(\root{12}{b^{12}}\) is \(b\).

Key Concepts

cube root

Understanding cube roots is fundamental when dealing with radical expressions. A cube root of a number is a value that, when raised to the power of three, gives the original number. For instance, the cube root of 8 is 2 because \(2^3 = 8\). Similarly, the cube root of \(a^3\) is \(a\). This property is represented as \(\sqrt[3]{a^3} = a\).
To put it simply, whenever you have a cube root of a perfect cube (like \(a^3\)), just strip away the cube root and the exponent!

twelfth root

The twelfth root follows the same principle but with the number 12. The twelfth root of a number is the value that, when raised to the power of twelve, gives the original number. For example, the twelfth root of \(b^{12}\) simplifies directly to \(b\). This property can be written as \(\sqrt[12]{b^{12}} = b\).
This works because \(b^{12}\) is a perfect power of twelve, making it straightforward to simplify using the properties of radicals.

properties of radicals

Radicals follow specific properties that make simplifying expressions easier. One key property to remember is \(\sqrt[n]{a^n} = a\) when n is the index of the root. This is especially useful for simplifying expressions like cube roots and twelfth roots.
Here are a few more useful properties:

• \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\) - You can split the root over multiplication.
• \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}\) - You can split the root over division.
• \((\sqrt[n]{a})^m = \sqrt[n]{a^m}\) - You can raise a radical to a power.

Using these properties effectively can simplify many complex radical expressions, making them easier to work with and understand.