To simplify the expression involving the square root of 2 multiplied by the cube root of 2, we can start by rewriting these roots in exponent form:
- The square root of 2 can be expressed as 2^(1/2).
- The cube root of 2 can be expressed as 2^(1/3).
Now we can combine these two parts:
2^(1/2) * 2^(1/3).
When multiplying expressions with the same base, we can add the exponents:
2^(1/2 + 1/3).
To add the fractions 1/2 and 1/3, we need a common denominator:
The least common denominator (LCD) for 2 and 3 is 6. Thus, we convert the fractions:
- 1/2 = 3/6
- 1/3 = 2/6
Now we can add them:
3/6 + 2/6 = 5/6.
So, our exponent becomes:
2^(5/6).
This is the simplified form of the original expression. Therefore, the simplified result of the square root of 2 multiplied by the cube root of 2 is:
2^(5/6).
You can also express this in radical form:
√[6]{2^5}, which means the sixth root of 2 raised to the power of 5.
In summary, both 2^(5/6) and √[6]{32} are equivalent representations of the simplified expression involving the square root of 2 multiplied by the cube root of 2.