Simplify the following radical expression. cube root of 27 x cubed end root
The Correct Answer and Explanation is:
Correct Answer: 3x
Explanation:
We are asked to simplify the cube root of the expression:
√[3]{27x³}
This is read as “the cube root of 27 times x cubed.”
Let us first break this expression into parts and simplify each part separately:
Step 1: Cube root of 27
27 is a perfect cube because:
3 × 3 × 3 = 27
So, the cube root of 27 is:
√[3]{27} = 3
Step 2: Cube root of x³
x³ means x raised to the power of 3. Taking the cube root of x³ is the same as reversing that power:
√[3]{x³} = x
This is because the cube root and the exponent of 3 cancel each other out.
Step 3: Combine the results
Now multiply the simplified cube root of each part:
√[3]{27x³} = √[3]{27} × √[3]{x³}
= 3 × x
= 3x
Additional Explanation:
Radical expressions involve roots, and in this case, we are dealing with cube roots. The cube root of a number asks, “What number multiplied by itself three times gives the original number?” Since 3 × 3 × 3 equals 27, the cube root of 27 is 3.
For variables, exponents and roots work in an opposite manner. If a variable is raised to a power that matches the root, they cancel out. In this case, since x³ is raised to the power of 3 and we are taking a cube root, we are left with x.
This concept is very helpful when simplifying expressions or solving equations involving radicals, especially in algebra and higher-level math. Simplifying expressions like this helps make the problem easier to solve or evaluate.
