Use the properties of exponents to perform the following operation:

Question

2^1/3 *8^1/4

Answer 1

To perform this operation using the properties of exponents, we can simplify each term separately and then multiply them together.

First, let's simplify 2^(1/3).

The exponent 1/3 means taking the cube root of 2. In other words, we are finding the number that, when multiplied by itself three times, equals 2.

To simplify 2^(1/3), we can rewrite it as the cube root of 2: ∛2.

Next, let's simplify 8^(1/4).

The exponent 1/4 means taking the fourth root of 8. In other words, we are finding the number that, when multiplied by itself four times, equals 8.

To simplify 8^(1/4), we can rewrite it as the fourth root of 8: ∜8.

Now, we can multiply the simplified terms together:
∛2 * ∜8

To multiply these terms, we need to simplify the radicands (the numbers inside the roots).

The cube root of 2 (∛2) cannot be simplified any further.

The fourth root of 8 (∜8) can be simplified. We can rewrite 8 as 2^3, so ∜8 = ∜(2^3) = ∜2^3.

Using the property of exponents that states ∜a^b = (a^(1/b))^b, we can rewrite the expression as:
(∜2^3)^1 = (2^(3/4))^1 = 2^(3/4).

Therefore, the result of 2^(1/3) * 8^(1/4) is 2^(3/4).