Question

Which of the following expressions is equivalent to 3√216/27^1/3 when applying the exponent properties?(1 point)

Options
1. 8
2. 1/2
3. 2
4. 18

Answer 1

[AI-generated answer — Bot GPT 3.5]

To simplify this expression using exponent properties, we need to rewrite the numbers with the same base.

3√216 can be written as (216)^(1/3).

27^1/3 can be written as (3^3)^(1/3).

Using the exponent rule (a^m)^n = a^mn, we can simplify (3^3)^(1/3) to 3^(3*(1/3)), which becomes 3^1, which is equal to 3.

Now, we can rewrite the expression as (216)^(1/3) / 3.

We can rewrite 216 as 6^3, so the expression becomes (6^3)^(1/3) / 3.

Using the exponent rule (a^m)^n = a^mn, we can simplify (6^3)^(1/3) to 6^(3*(1/3)), which becomes 6^1, which is equal to 6.

Now, the expression becomes 6 / 3, which simplifies to 2.

Therefore, option 3. 2 is the correct answer.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the equivalent expression using the exponent properties, we need to apply the rules of exponents.

First, let's simplify the expression 27^(1/3):

27^(1/3) can be written as the cube root of 27.

The cube root of 27 is 3, since 3 multiplied by itself three times equals 27.

So, 27^(1/3) simplifies to 3.

Now, let's substitute this result into the original expression:

3√216/27^(1/3) becomes 3√216/3.

Next, let's simplify the expression inside the cube root:

The cube root of 216 can be found by finding the cube root of its prime factor, 216 = 2 * 2 * 2 * 3 * 3 * 3 = 2^3 * 3^3.

The cube root of 2^3 is 2, and the cube root of 3^3 is 3.

Therefore, the cube root of 216 is equal to 2 * 3 = 6.

Substituting this result into the expression gives us:

3√216/3 = 6/3 = 2.

So, the equivalent expression is 2.

Therefore, the correct answer is option 3.