find the product: (4k to the 5th power)(-2k)to the 3rd power. 5th power is inside the parenthesis and the 3rd is outside the parenthesis.

Is it -32k to the 8th power or -32k to the 2nd power?

(4k^5)(-2k)^3

4^5 k^5 (-2)^3 k^3
1024(-8) k^8
-8196 k^8

Now, if you meant
4(k^5)(-2k)^3
Then you do indeed have
-32k^8

When you have minus signs in the coefficient, they do not affect the exponents. Multiplying powers means add exponents. To get k^2 = k^(5-3), you'd have to have

4k^5(-2k)^-3
= -4/8 k^2

Then you'd have to take a negative power of 2, as well, making it a fraction.

To find the product, we need to apply the exponent rules.

First, let's simplify the expression inside the parentheses:
(-2k)^3 = (-2)^3 * k^3 = -8 * k^3

Now, let's multiply that result by (4k)^5:
(4k)^5 * (-8k^3) = 4^5 * k^5 * (-8) * k^3 = 1024 * k^5 * (-8) * k^3 = -8192 * k^8

So, the product is -8192k^8.

To find the product, let's break it down step by step.

First, let's simplify the expression inside the parentheses, which is (4k)^5. To raise a power to another power, we multiply the exponents, so (4k)^5 becomes 4^5 * k^5. This gives us (1024 * k^5).

Next, let's simplify the expression outside the parentheses, which is (-2k)^3. Again, to raise a power to another power, we multiply the exponents. So (-2k)^3 becomes (-2)^3 * k^3, which is equal to (-8 * k^3).

Now, let's multiply the two terms together: (1024 * k^5) * (-8 * k^3).

To multiply these terms, we multiply the coefficients (numbers) and the variables separately. The product of the coefficients is (1024 * -8), which is -8192.

For the variables, since both terms have 'k', we add the exponents. So, k^5 * k^3 becomes k^(5+3), which is k^8.

Putting it all together, the final product is -8192 * k^8. Therefore, the answer is -8192k to the 8th power.