Solve for n -646=-3(65-n)^(3/2)+2
The given problem is an algebraic equation where you are asked to find the value of the variable n. The equation is -646 = -3(65-n)^(3/2) + 2, and it involves an operation with an exponent. Specifically, the term (65-n)^(3/2) suggests that you need to work with a fractional exponent or a radical (in this case, a square root raised to the third power). The equation requires you to isolate and solve for n by performing inverse operations and properly handling the exponent to find the value that makes the equation true.
Transform the given equation to
Isolate the terms involving
Subtract 2 from both sides to get
Combine the constants on the right-hand side, yielding
Eliminate the fractional exponent by raising both sides to the power of
Simplify the equation.
Work on the left side:
Apply the product rule to
Multiply the exponents in
Use the power rule:
Simplify by canceling out the common factor of 3.
Further simplify by canceling out the common factor of 2.
The expression simplifies to
Distribute
Reorder the terms for clarity.
Determine the value of
Isolate the term with
Divide by
Divide each term to isolate
Simplify the left side by canceling out the common factor.
Simplify the right side by working through each term.
Apply the power of quotient rule and simplify.
Add the constants to find the value of
This problem involves several key algebraic concepts:
Isolating Variables: Moving all terms containing the variable of interest to one side of the equation to facilitate solving.
Fractional Exponents: Understanding that
Power Rules: Applying the rule
Distributive Property: Multiplying a single term by each term within a set of parentheses.
Simplifying Expressions: Reducing expressions by combining like terms and canceling common factors.
Negative Numbers: Handling operations with negative numbers, particularly when dividing or multiplying them.
Cube Roots and Square Roots: Recognizing that raising to the power of