Combine and simplify the following radical expression. [tex]$\frac{2}{3 \sqrt[3]{5}}$[/tex]
Answer (2)
To simplify 3 3 5 2 , multiply both numerator and denominator by 3 25 to eliminate the cube root in the denominator. This results in the simplified expression of 15 2 3 25 . The final answer is 15 2 3 25 .
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Multiply the numerator and denominator by 3 5 2 to rationalize the denominator.
Simplify the denominator: 3 3 5 ⋅ 3 5 2 = 3 3 5 3 = 3 ⋅ 5 = 15 .
The simplified expression is 15 2 3 25 .
The final answer is 15 2 3 25 .
Explanation
Understanding the Problem We are given the expression 3 3 5 2 and our goal is to simplify it by rationalizing the denominator.
Rationalizing the Denominator To rationalize the denominator, we need to get rid of the cube root. We can do this by multiplying both the numerator and the denominator by a factor that will make the exponent of 5 in the denominator an integer. Since we have 3 5 in the denominator, we can multiply by 3 5 2 to get 3 5 3 = 5 .
Multiplying by the Rationalizing Factor Multiply both the numerator and the denominator by 3 5 2 = 3 25 : 3 3 5 2 ⋅ 3 5 2 3 5 2 = 3 3 5 3 25 2 3 25 = 3 3 5 3 2 3 25
Simplifying the Expression Now, simplify the denominator: 3 3 5 3 2 3 25 = 3 ⋅ 5 2 3 25 = 15 2 3 25
Final Answer The simplified expression is 15 2 3 25 .
Examples
Imagine you are a chef and a recipe calls for a specific amount of a rare spice, but your measuring spoons are not precise enough for cube roots. Rationalizing the denominator helps you convert the recipe to use more easily measured quantities while keeping the ratio of ingredients the same. This technique is useful in any situation where you need to work with radicals and want to express them in a more convenient form, such as in engineering, physics, or computer graphics.
