Problem

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((65n))32+2

Answer

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Solution:

Step:1

Transform the given equation to 3(65n)32+2=646.

Step:2

Isolate the terms involving n on one side.

Step:2.1

Subtract 2 from both sides to get 3(65n)32=648.

Step:2.2

Combine the constants on the right-hand side, yielding 3(65n)32=648.

Step:3

Eliminate the fractional exponent by raising both sides to the power of 23: ((3(65n)32))23=(648)23.

Step:4

Simplify the equation.

Step:4.1

Work on the left side: ((3))23((65n)32)23=(648)23.

Step:4.1.1

Apply the product rule to 3(65n)32.

Step:4.1.2

Multiply the exponents in ((65n)32)23.

Step:4.1.2.1

Use the power rule: ((3))23(65n)3223=(648)23.

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))23(65n)=(648)23.

Step:4.1.4

Distribute ((3))23 across 65n.

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))23 from both sides.

Step:5.2

Divide by ((3))23 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 amn is equivalent to the nth root of am.

  3. Power Rules: Applying the rule (am)n=amn and the quotient rule aman=amn.

  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 23 is equivalent to taking the cube root and then squaring the result.

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