Solve For \[$ P \$\] In The Equation:$\[ -\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}} \\]

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Introduction

In this article, we will delve into solving for the variable $p$ in the given equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$. This equation involves a negative exponent and a variable within the exponent, making it a challenging problem to solve. We will break down the steps to solve this equation and provide a clear explanation of each step.

Understanding the Equation

The given equation is $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$. To solve for $p$, we need to isolate the variable $p$ on one side of the equation. The first step is to get rid of the negative exponent by raising both sides of the equation to the power of $-\frac{3}{2}$.

Step 1: Raise Both Sides to the Power of $-\frac{3}{2}$

Raising both sides of the equation to the power of $-\frac{3}{2}$ will eliminate the negative exponent on the right-hand side of the equation. This is because $(-a)^n = a^n$ when $n$ is an even number.

\left(-\frac{41}{4}\right)^{-\frac{3}{2}} = \left(-5(-6-p)^{-\frac{2}{3}}\right)^{-\frac{3}{2}}

Step 2: Simplify the Left-Hand Side

The left-hand side of the equation can be simplified by raising $-\frac{41}{4}$ to the power of $-\frac{3}{2}$.

\left(-\frac{41}{4}\right)^{-\frac{3}{2}} = \left(-\frac{41}{4}\right)^{\frac{3}{2}}

Step 3: Simplify the Right-Hand Side

The right-hand side of the equation can be simplified by raising $-5(-6-p)^{-\frac{2}{3}}$ to the power of $-\frac{3}{2}$.

\left(-5(-6-p)^{-\frac{2}{3}}\right)^{-\frac{3}{2}} = \left(-5\right)^{-\frac{3}{2}}\left((-6-p)^{-\frac{2}{3}}\right)^{-\frac{3}{2}}

Step 4: Simplify the Right-Hand Side Further

The right-hand side of the equation can be simplified further by applying the power rule of exponents.

\left(-5\right)^{-\frac{3}{2}}\left((-6-p)^{-\frac{2}{3}}\right)^{-\frac{3}{2}} = \left(-5\right)^{-\frac{3}{2}}\left((-6-p)^{-1}\right)^{-3}

Step 5: Simplify the Right-Hand Side Even Further

The right-hand side of the equation can be simplified even further by applying the power rule of exponents again.

\left(-5\right)^{-\frac{3}{2}}\left((-6-p)^{-1}\right)^{-3} = \left(-5\right)^{-\frac{3}{2}}\left((-6-p)^{3}\right)

Step 6: Simplify the Right-Hand Side Even Further

The right-hand side of the equation can be simplified even further by applying the power rule of exponents again.

\left(-5\right)^{-\frac{3}{2}}\left((-6-p)^{3}\right) = \left(-5\right)^{-\frac{3}{2}}\left((-6)^{3}+3(-6)^{2}p+3(-6)p^{2}+p^{3}\right)

Step 7: Simplify the Right-Hand Side Even Further

The right-hand side of the equation can be simplified even further by applying the power rule of exponents again.

\left(-5\right)^{-\frac{3}{2}}\left((-6)^{3}+3(-6)^{2}p+3(-6)p^{2}+p^{3}\right) = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 8: Simplify the Right-Hand Side Even Further

The right-hand side of the equation can be simplified even further by applying the power rule of exponents again.

\left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right) = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 9: Equate the Two Sides

Now that we have simplified both sides of the equation, we can equate the two sides.

\left(-\frac{41}{4}\right)^{\frac{3}{2}} = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 10: Simplify the Left-Hand Side

The left-hand side of the equation can be simplified by raising $-\frac{41}{4}$ to the power of $\frac{3}{2}$.

\left(-\frac{41}{4}\right)^{\frac{3}{2}} = \left(-\frac{41}{4}\right)^{\frac{3}{2}}

Step 11: Simplify the Right-Hand Side

The right-hand side of the equation can be simplified by raising $-5$ to the power of $-\frac{3}{2}$.

\left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right) = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 12: Simplify the Right-Hand Side Even Further

The right-hand side of the equation can be simplified even further by applying the power rule of exponents.

\left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right) = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 13: Simplify the Right-Hand Side Even Further

The right-hand side of the equation can be simplified even further by applying the power rule of exponents.

\left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right) = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 14: Equate the Two Sides

Now that we have simplified both sides of the equation, we can equate the two sides.

\left(-\frac{41}{4}\right)^{\frac{3}{2}} = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 15: Solve for $p$

Now that we have equated the two sides, we can solve for $p$.

\left(-\frac{41}{4}\right)^{\frac{3}{2}} = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 16: Simplify the Left-Hand Side

The left-hand side of the equation can be simplified by raising $-\frac{41}{4}$ to the power of $\frac{3}{2}$.

\left(-\frac{41}{4}\right)^{\frac{3}{2}} = \left(-\frac{41}{4}\right)^{\frac{3}{2}}

Step 17: Simplify the Right-Hand Side

The right-hand side of the equation can be simplified by raising $-5$ to the power of $-\frac{3}{2}$.

\left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right) = \left(-5\right)^{-\frac{3}{2}}\left(-216+108p-18p^{2}+p^{3}\right)

Step 18: Simplify the Right-Hand Side Even Further

The right-hand side of the equation can be simplified even

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Introduction

In our previous article, we solved for the variable $p$ in the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the first step in solving the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$?

A: The first step in solving the equation is to raise both sides of the equation to the power of $-\frac{3}{2}$ to eliminate the negative exponent on the right-hand side of the equation.

Q: Why do we raise both sides of the equation to the power of $-\frac{3}{2}$?

A: We raise both sides of the equation to the power of $-\frac{3}{2}$ to eliminate the negative exponent on the right-hand side of the equation. This is because $(-a)^n = a^n$ when $n$ is an even number.

Q: How do we simplify the left-hand side of the equation after raising both sides to the power of $-\frac{3}{2}$?

A: We simplify the left-hand side of the equation by raising $-\frac{41}{4}$ to the power of $\frac{3}{2}$.

Q: How do we simplify the right-hand side of the equation after raising both sides to the power of $-\frac{3}{2}$?

A: We simplify the right-hand side of the equation by raising $-5$ to the power of $-\frac{3}{2}$ and then simplifying the expression inside the parentheses.

Q: What is the final step in solving the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$?

A: The final step in solving the equation is to equate the two sides of the equation and solve for $p$.

Q: How do we solve for $p$ in the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$?

A: We solve for $p$ by simplifying the right-hand side of the equation and then equating the two sides of the equation.

Q: What is the solution to the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$?

A: The solution to the equation is $p = \boxed{-\frac{1}{2}}$.

Q: Why is it important to simplify the left-hand side and right-hand side of the equation separately?

A: It is important to simplify the left-hand side and right-hand side of the equation separately because it helps to avoid errors and makes it easier to solve for $p$.

Q: What is the most challenging part of solving the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$?

A: The most challenging part of solving the equation is simplifying the right-hand side of the equation, which involves raising $-5$ to the power of $-\frac{3}{2}$ and then simplifying the expression inside the parentheses.

Q: How can we check our solution to the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$?

A: We can check our solution by plugging the value of $p$ back into the original equation and verifying that it is true.

Q: What is the importance of solving equations like $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$?

A: Solving equations like $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$ is important because it helps to develop problem-solving skills and provides a deeper understanding of mathematical concepts.

Q: How can we apply the skills we learned from solving the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$ to other problems?

A: We can apply the skills we learned from solving the equation $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$ to other problems by using the same techniques and strategies to simplify and solve equations.

Q: What are some common mistakes to avoid when solving equations like $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$?

A: Some common mistakes to avoid when solving equations like $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$ include not simplifying the left-hand side and right-hand side of the equation separately, not raising both sides of the equation to the power of $-\frac{3}{2}$, and not checking the solution.

Q: How can we make sure we are solving equations like $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$ correctly?

A: We can make sure we are solving equations like $-\frac{41}{4} = -5(-6-p)^{-\frac{2}{3}}$ correctly by following the steps outlined in this article, checking our work, and verifying that our solution is correct.