Which expression is equivalent to the cube root of 216x^27?
The correct answer and explanation is:
Correct Answer:
The expression equivalent to the cube root of 216x27216x^{27} is: 216×273=6×9\sqrt[3]{216x^{27}} = 6x^9
Explanation (Approximately 300 Words):
To simplify the cube root of an expression like 216×273\sqrt[3]{216x^{27}}, we apply the properties of radicals and exponents.
Step 1: Factor the expression inside the radical
Start with: 216×273\sqrt[3]{216x^{27}}
We break this down into two parts: the cube root of the number and the cube root of the variable part: 2163⋅x273\sqrt[3]{216} \cdot \sqrt[3]{x^{27}}
Step 2: Simplify the cube root of the number
216216 is a perfect cube. We can find this by testing small integers: 63=6⋅6⋅6=2166^3 = 6 \cdot 6 \cdot 6 = 216
So: 2163=6\sqrt[3]{216} = 6
Step 3: Simplify the cube root of the variable expression
Use the rule of exponents under radicals: x273=x27÷3=x9\sqrt[3]{x^{27}} = x^{27 \div 3} = x^9
Step 4: Multiply the simplified parts
Now that both parts are simplified: 216×273=6⋅x9=6×9\sqrt[3]{216x^{27}} = 6 \cdot x^9 = 6x^9
Key Concepts Used:
- Cube root of a product: ab3=a3⋅b3\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}
- Cube root of a power: xn3=xn/3\sqrt[3]{x^n} = x^{n/3}, provided nn is divisible by 3
- Recognizing perfect cubes: 13=1,23=8,33=27,43=64,53=125,63=2161^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125, 6^3 = 216
Final Answer:
6×9\boxed{6x^9}
This simplified expression is much easier to work with in algebra and calculus and represents the exact same value as the original radical expression.