Question

1) (-27)^1/3= 2) (-27)^-1/3= 3) 27^-1/3= 4) 27^1/3 * 8^1/3=

Answer 1

1) ∛(-27)=-3             х^(1/3)=∛х;   х^(-1/3)=1/x^(1/3)=1/∛x

2) 1/∛(-27)=-1/3

3) 1/∛27=1/3

4) ∛27 * ∛8=3*2=6.

Answer 2

To simplify these expressions, we need to understand the rules of exponents and roots.

1. (-27)^(1/3): The cube root of a negative number will have one real root and two complex roots. The principal (real) cube root of -27 is -3, because (-3)^3 = -27. So, (-27)^(1/3) = -3.

2. (-27)^(-1/3): To calculate this, we can take the reciprocal of the cube root of -27. The reciprocal of -3 is -1/3. So, (-27)^(-1/3) = -1/3.

3. 27^(-1/3): The cube root of 27 is 3, because 3^3 = 27. To find the reciprocal, we take the inverse of 3, which is 1/3. So, 27^(-1/3) = 1/3.

4. 27^(1/3) * 8^(1/3): We can simplify this expression by calculating each cube root separately and then multiplying them.

• The cube root of 27 is 3.
• The cube root of 8 is 2, because 2^3 = 8.

So, 27^(1/3) * 8^(1/3) = 3 * 2 = 6.

Therefore, the results are:

1. (-27)^(1/3) = -3
2. (-27)^(-1/3) = -1/3
3. 27^(-1/3) = 1/3
4. 27^(1/3) * 8^(1/3) = 6