Final answer:
Only Location 3: 20x^4√7x + 8x^4√7x matches the simplified form of the original expression 5x√28^5 + 8x^3√7x when x > 0. This is identified by simplifying the given expression and matching it with the options provided.
Step-by-step explanation:
We are asked to find which expressions are equivalent to the given expression 5x√28^5 + 8x^3√7x, assuming x > 0. To solve this, we have to simplify and manipulate each expression to see if it matches with our given expression.
Firstly, 5x√28^5 can be simplified. The √28 can be written as 28^0.5, and raising it to the power of 5 gives us 28^2.5, which is 28^2 * 28^0.5, or (28×28)√28, which simplifies to 784√28. Multiplying 784 by 5x gives 3920x√28.
Next, we look at 8x^3√7x. We know that x^0.5 is the same as √x, hence, √7x is 7^0.5 * x^0.5. That gives us 7^0.5 × x^(3+0.5), which simplifies to 7^0.5 × x^3.5. Multiplying by 8 gives us 8x^3.5√7.
We can now combine these two pieces to get the simplified form of the original expression: 3920x√28 + 8x^3.5√7.
When we examine all the listed locations:
- Location 1 doesn't match our simplified expression.
- Location 2 doesn't either, as the structure doesn't resemble our simplified expression.
- Location 3 is our original expression as it stands, so it can be considered equivalent.
- Location 4 simplifies differently and thus is not equivalent.
- Location 5, after combining like terms, gives 18x^3√7x, which does not match.
- Location 6 directly lists 18x^3√7x, which is a portion of our simplified expression but does not match the entire expression.
- Therefore, the correct location providing an equivalent expression is Location 3 only.