Can the equation x^3=−217 yield a positive final solution? Why or why not?
No, because all cubed numbers yield negative solutions.
No, because you must cube a negative number to get a negative answer.
Yes, because multiplying three negatives together will equal a positive number.
Yes, because cubed roots can give two solutions.
No, because all cubed numbers yield negative solutions.
To determine if the equation x^3=−217 can yield a positive final solution, we need to consider the properties of exponents and the nature of cube roots.
First, let's examine the equation x^3 = −217. The left side of the equation represents x raised to the power of 3, while the right side is a negative number.
To find the value of x, we can take the cube root of both sides of the equation. The cube root (denoted as ∛) of a number is a value that, when raised to the power of 3, gives the original number.
However, it's important to note that taking the cube root of a number can give two possible solutions since both positive and negative numbers can be cubed to obtain the same result. So, when we solve x^3 = −217, we'll find two potential values for x.
∛( x^3 ) = ∛(−217)
Now let's analyze the statements given:
1. "No, because all cubed numbers yield negative solutions."
This statement is incorrect because cubing a number can yield both positive and negative solutions as mentioned before.
2. "No, because you must cube a negative number to get a negative answer."
This statement is also incorrect. While it's true that multiplying three negative numbers together will give a negative result, this doesn't mean that the only solution to x^3 = −217 is a negative number.
3. "Yes, because multiplying three negatives together will equal a positive number."
This statement is incorrect because the equation x^3 = −217 does not involve multiplying three negatives together. Instead, it's a single variable equation representing the cube of x.
4. "Yes, because cubed roots can give two solutions."
This statement is correct. Since cube roots can yield two solutions, it is possible that x^3 = −217 has a positive final solution along with a negative one. To confirm this, we need to calculate the cube root of −217.
∛(−217) is approximately equal to -6.365.
Therefore, the equation x^3 = −217 can have a negative final solution (-6.365) but not a positive one since all cubed roots of a negative number are themselves negative.
Yes, because cubed roots can give two solutions.
To solve the equation x^3 = -217, we can take the cubed root of both sides of the equation:
∛(x^3) = ∛(-217)
Taking the cubed root of x^3 gives us:
x = ∛(-217)
The cubed root of a negative number can yield both a positive and a negative solution. Therefore, it is possible for this equation to yield a positive final solution.