Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents. 4x3)(2x−1)3/(14y3)(4y−2)
To simplify the expression, we can use the properties of exponents and apply them step by step.
First, let's simplify each factor individually.
In 4x^3, we have a numerical factor of 4 and a variable factor of x raised to the power of 3. We can write this as 4x^3 = 4 * x^3.
In (2x^(-1))^3, we have a numerical factor of 2 and a variable factor of x raised to the power of -1, which means it is in the denominator. When we raise a fraction to a power, we raise both the numerator and the denominator to that power. Applying this, we get (2x^(-1))^3 = (2^3 * x^(-1)^3) = 8x^(-3).
In (14y^3), we have a numerical factor of 14 and a variable factor of y raised to the power of 3. We can write this as 14y^3 = 14 * y^3.
In (4y^(-2)), we have a numerical factor of 4 and a variable factor of y raised to the power of -2, which means it is in the denominator. Similarly, we can rewrite this as (4y^(-2)) = 4 * y^(-2).
Now, let's substitute these simplified factors back into the original expression:
(4x^3)(2x^(-1))^3 / (14y^3)(4y^(-2))
= (4 * x^3)(8x^(-3)) / (14 * y^3)(4 * y^(-2))
= (4 * 8 * x^3 * x^(-3)) / (14 * 4 * y^3 * y^(-2))
Multiplying the numerical factors, we get:
32 * x^3 * x^(-3) / 56 * y^3 * y^(-2)
Using the properties of exponents, we can simplify the variables by adding the exponents:
= 32 * x^(3 + (-3)) / 56 * y^(3 + (-2))
= 32 * x^0 / 56 * y^1
Any number raised to the power of 0 is equal to 1, so x^0 = 1. Therefore, we have:
= 32 * 1 / 56 * y
= 32 / 56 * y
Finally, we can simplify the fraction by dividing both the numerator and denominator by their greatest common factor, which is 8:
= 4 / 7 * y
So, the simplified expression using the properties of exponents is 4/7y.