Question

Combine radicals, if possible. \(8 \sqrt[3]{3}-2 \sqrt[3]{3}-2 \sqrt{3}\)

Short answer

Question: Simplify the expression \(8 \sqrt[3]{3} - 2 \sqrt[3]{3} - 2 \sqrt{3}\). Answer: The simplified expression is \(6 \sqrt[3]{3} - 2 \sqrt{3}\).

Step-by-step solution

1. Identify the like radicals

The expression contains two cube root terms (\(8 \sqrt[3]{3}\) and \(-2 \sqrt[3]{3}\)) and one square root term (\(-2 \sqrt{3}\)). Since the radicals are different for these terms, we can only combine the cube root terms.

2. Combine the cube root terms

We can combine the cube root terms by applying the properties of radicals. \(8 \sqrt[3]{3} - 2 \sqrt[3]{3} = (8 - 2) \sqrt[3]{3}\). We will now subtract the coefficients to get the simplified cube root term.

3. Calculate the simplified cube root term

\((8 - 2) \sqrt[3]{3} = 6 \sqrt[3]{3}\). Now, we have the simplified cube root term.

4. Rewrite the expression with the simplified cube root term

Replacing the original cube root terms with the simplified term and including the square root term, the final expression is: \(6 \sqrt[3]{3} - 2 \sqrt{3}\). The simplified expression is \(6 \sqrt[3]{3} - 2 \sqrt{3}\).

Key concepts

Cube Roots

Cube roots are essential in mathematics as they represent one of the fundamental roots. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because

• 2 × 2 × 2 = 8.

The cube root is commonly notated as

• the radical symbol with an index of 3: \(\sqrt[3]{x}.\)

When dealing with cube roots, it's crucial to understand how to simplify and combine them. Like other radicals, cube roots can be manipulated using property rules,but they have to share the same radicand (the number inside the radical) to be combined. So, expressions like \(8 \sqrt[3]{3} - 2 \sqrt[3]{3}\) can be combined as they both have the same radicand of 3.This combination works just like combining like terms in algebra:

• Subtract the coefficients: \(8 - 2 = 6\).

Ultimately, this results in \(6 \sqrt[3]{3}\).

Square Roots

Square roots function similarly to cube roots but focus on repeated multiplication of a number by itself twice. The square root of a number \(x\) is another number \(y\) such that

• \(y \times y = x.\)

For instance, the square root of 9 is 3 (because 3 × 3 = 9).The square root is symbolized by the radical sign \(\sqrt{}\).When working with square roots, it's essential to discern when terms can be combined. Like cube roots, they must share the same radicand. In our original exercise, \(-2 \sqrt{3}\) stands alone since it shares no commonality in its radicand with the cube root terms.Even though both cube and square roots fall under the category of radicals, they cannot be combined unless they have the same type of root and identical radicands. Therefore, in expressions with mixed radicals – like cube roots and square roots, as seen in our exercise – each type remains separate.

Properties of Radicals

Understanding the properties of radicals is crucial when simplifying and combining radical expressions. These properties allow us to perform algebraic operations more efficiently. Here are some fundamental properties:

• Multiplication Property: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab},\)which means you can multiply the numbers under the radical sign directly.
• Division Property: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}.\)This property is useful when simplifying fractions that are under a radical sign.
• Like Radicals: Only radicals with the same index and radicand can be combined. For example, \(8 \sqrt[3]{3} - 2 \sqrt[3]{3}\) can be simplified to \(6 \sqrt[3]{3}\).

Remember, these properties of radicals help maintain the consistency of operations and ensure that radical expressions remain simplified. When in doubt, always check if the radicals you're working with share identical indices and radicands before attempting to combine or simplify them. These properties ground our understanding and manipulation of complex expressions in algebra and beyond.