Solve for n -646=-3(65-n)^(3/2)+2

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:

1. Isolating Variables: Moving all terms containing the variable of interest to one side of the equation to facilitate solving.

2. Fractional Exponents: Understanding that $a^{\frac{m}{n}}$ is equivalent to the nth root of $a^m$.

3. Power Rules: Applying the rule $(a^m{)}^n=a^{mn}$ and the quotient rule $\frac{a^m}{a^n}=a^{m-n}$.

4. Distributive Property: Multiplying a single term by each term within a set of parentheses.

5. Simplifying Expressions: Reducing expressions by combining like terms and canceling common factors.

6. Negative Numbers: Handling operations with negative numbers, particularly when dividing or multiplying them.

7. 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.