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.
$- 646 = - 3 \left(\left(\right. 65 - n \left.\right)\right)^{\frac{3}{2}} + 2$
Answer
Solution:
Step:1
Transform the given equation to $-3(65 - n)^{\frac{3}{2}} + 2 = -646$.
Step:2
Isolate the terms involving $n$ on one side.
Step:2.1
Subtract 2 from both sides to get $-3(65 - n)^{\frac{3}{2}} = -648$.
Step:2.2
Combine the constants on the right-hand side, yielding $-3(65 - n)^{\frac{3}{2}} = -648$.
Step:3
Eliminate the fractional exponent by raising both sides to the power of $\frac{2}{3}$: $((-3(65 - n)^{\frac{3}{2}}))^{\frac{2}{3}} = (-648)^{\frac{2}{3}}$.
Step:4
Simplify the equation.
Step:4.1
Work on the left side: $((-3))^{\frac{2}{3}}((65 - n)^{\frac{3}{2}})^{\frac{2}{3}} = (-648)^{\frac{2}{3}}$.
Step:4.1.1
Apply the product rule to $-3(65 - n)^{\frac{3}{2}}$.
Step:4.1.2
Multiply the exponents in $((65 - n)^{\frac{3}{2}})^{\frac{2}{3}}$.
Step:4.1.2.1
Use the power rule: $((-3))^{\frac{2}{3}}(65 - n)^{\frac{3}{2} \cdot \frac{2}{3}} = (-648)^{\frac{2}{3}}$.
Step:4.1.2.2
Simplify by canceling out the common factor of 3.
Step:4.1.2.3
Further simplify by canceling out the common factor of 2.
Step:4.1.3
The expression simplifies to $((-3))^{\frac{2}{3}}(65 - n) = (-648)^{\frac{2}{3}}$.
Step:4.1.4
Distribute $((-3))^{\frac{2}{3}}$ across $65 - n$.
Step:4.1.5
Reorder the terms for clarity.
Step:5
Determine the value of $n$.
Step:5.1
Isolate the term with $n$ by subtracting $65((-3))^{\frac{2}{3}}$ from both sides.
Step:5.2
Divide by $-((-3))^{\frac{2}{3}}$ to solve for $n$.
Step:5.2.1
Divide each term to isolate $n$.
Step:5.2.2
Simplify the left side by canceling out the common factor.
Step:5.2.3
Simplify the right side by working through each term.
Step:5.2.3.1
Apply the power of quotient rule and simplify.
Step:5.2.3.2
Add the constants to find the value of $n$: $n = 29$.
Knowledge Notes:
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 $a^{\frac{m}{n}}$ is equivalent to the nth root of $a^m$.
Power Rules: Applying the rule $(a^m)^n = a^{mn}$ and the quotient rule $\frac{a^m}{a^n} = a^{m-n}$.
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 $\frac{2}{3}$ is equivalent to taking the cube root and then squaring the result.
