Multiply (8x^3)(2x)^-3. What is true about (8x^3) and (2x)^-3?

Question

The only other thing that my text book says is (Hint: Simplify the expression.). I forgot how to solve with a negative exponent and I can't find the solution in my text book. (2x)^-3 is really starting to confuse me. Then, it wants me to find out what is true about (8x^3) and (2x)^-3 and I don't even understand that.

Answer 1

Big hint: Remember that you add exponents when multiplying like expressions---eg x^3 * x^4 = x^7

If you add exponents to multiply then you subtract exponents to divide....

So:

You are given: (8x^3)*(2x)^-3

This is the same as: (8x^3) / (2x)^3

Because the 2x is being raised to a power, we can do the division yet. First: (2x)^3 = 8x^3

Now we can divide: (8x^3) / (8x^3) = 1

Note that we did not need to create a fraction. Example: suppose you are given x^6 * x^-4. The bases are the same so simply add the exponents: 6 + -4 = 2, so x^6 * x^-4 = x^2

Answer 2

IA negative exponent tells you to use the reciprocal of the value. For example (2)^-1 is equal to 1/2 while 2 ^1 is just 2.

For your problem (8x^3) and (2x)^-3 are reciprocals of one another. (2x)^-3 is equal to 1/((2x)^3. This is equal to 1/8x^3

Answer 3

What you need to understand is that x to the -n = 1/(x to the n).

So 8(x to the 3rd power) X (2x to the -3rd power) = (8)(x to the 3rd power) / (2x to the 3rd power)

= (8)(x to the 3rd power) / {(8)(x to the 3rd power)}

= 1

Therefore, the answer to the 2nd question is that they are reciprocals (multiplicative inverses). Just like x and 1/x, their product = 1.

Also, please note that the location of the parentheses is crucial in problems like these. The answer would be different had 8x been inside the parentheses and 3 outside, or had the -3 been inside the parentheses. Check the problem you were given to make sure it's exactly as you have it written here.