In Exercises 45–66, divide and, if possible, simplify.__³√54³√2

Verified step by step guidance

1. insert step 1: Recognize that you are dividing two cube roots: \( \frac{\sqrt[3]{54}}{\sqrt[3]{2}} \).
2. insert step 2: Use the property of radicals that allows you to combine the division under a single cube root: \( \sqrt[3]{\frac{54}{2}} \).
3. insert step 3: Simplify the fraction inside the cube root: \( \frac{54}{2} = 27 \).
4. insert step 4: Now, you have \( \sqrt[3]{27} \).
5. insert step 5: Recognize that 27 is a perfect cube, as \( 27 = 3^3 \), so \( \sqrt[3]{27} = 3 \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this case, we are dealing with cube roots, which are expressed as ³√x, meaning the number that, when multiplied by itself three times, gives x. Understanding how to manipulate these expressions is crucial for simplifying and dividing them.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form. This can include factoring out perfect cubes from under the radical sign. For example, ³√54 can be simplified by recognizing that 54 = 27 × 2, allowing us to extract the cube root of 27, which is 3.

Division of Radicals

Dividing radical expressions requires applying the property that states ³√a / ³√b = ³√(a/b). This means you can combine the radicands (the numbers inside the radical) into a single radical expression. This property is essential for simplifying the division of cube roots in the given problem.