Question

Evaluate 27^(-1/3) . 32^(-2/5). Express your answer as a common fraction.

Options:
a) 1/12

b) 1/36
​

c) 1/64
​

d) 1/108
​

Answer 1

To evaluate the expression 27^(-1/3) . 32^(-2/5), we can simplify each exponent separately and then perform the multiplication. Simplifying first, 27^(-1/3) becomes 1/3√1 / 3√27 and 32^(-2/5) becomes 1/3√2. Multiplying these simplified expressions, the final value is 1/3√2, equivalent to the fraction 1/√(18).

Step-by-step explanation:

To evaluate the expression 27^(-1/3) . 32^(-2/5), we can simplify each exponent separately and then perform the multiplication.

Let's simplify 27^(-1/3) first:

Remember that a negative exponent indicates the reciprocal of the base raised to the positive exponent. So, 27^(-1/3) is equal to (1/27)^(1/3).

Simplifying further, we have (1/27)^(1/3) = 1^1/3 / 27^(1/3) = 1/3√1 / 3√27.

Now let's simplify 32^(-2/5):

Using the same principle, we have 32^(-2/5) = (1/32)^(2/5).

Now we can multiply the simplified expressions: (1/3√1 / 3√27) . (1/32)^(2/5).

To simplify further, we need to rewrite 3√1 as 1 and 3√27 as 3.

So, the expression simplifies to 1/3 . (1/32)^(2/5) = 1/3 . (1/2)^(2/5) = 1/3 . 1/√2 = 1/3√2.

Therefore, the value of the expression 27^(-1/3) . 32^(-2/5) is 1/3√2, which is equivalent to the fraction 1/√(18).