Solving For A^p * A^q * A^r Given X = -(y + Z)
In this comprehensive article, we will delve into the step-by-step solution of a fascinating mathematical problem. Our primary focus will be on determining the value of a^p * a^q * a^r, given the initial condition that x = -(y + z), and the subsequent definitions of p, q, and r. This problem not only requires a solid understanding of algebraic manipulation and exponent rules but also offers an excellent opportunity to hone our problem-solving skills. The discussion will cover each stage in detail, ensuring clarity and ease of understanding for readers of all mathematical backgrounds. We will start by outlining the given equations and then methodically simplify and substitute values to arrive at the final answer. By breaking down the problem into manageable steps, we aim to provide a clear and concise solution that is both informative and accessible.
To begin, let's reiterate the given variables and equations that form the foundation of our problem. These definitions are crucial as they set the stage for all subsequent calculations and simplifications. We are given the following:
- x = -(y + z): This equation establishes a direct relationship between x, y, and z, indicating that x is the negative sum of y and z. This is a key piece of information that we will use later for substitutions and simplifications.
- p = x^5 / (x^3 * y * z): This defines p as a fraction involving x, y, and z, where the numerator is x raised to the power of 5, and the denominator is the product of x cubed, y, and z. Simplifying this expression will be a critical step in finding the value of a^p.
- q = y^5 / (x * y^3 * z): Similarly, q is defined as a fraction involving x, y, and z, with y raised to the power of 5 in the numerator, and the product of x, y cubed, and z in the denominator. Simplifying q will help us determine the value of a^q.
- r = z^2 / (x * y * z^3): Lastly, r is expressed as a fraction involving x, y, and z, with z squared in the numerator and the product of x, y, and z cubed in the denominator. Simplifying r will be essential for calculating the value of a^r.
Understanding these variables and their relationships is the first step in solving the problem. We will now proceed to simplify each of the expressions for p, q, and r, making use of algebraic rules and the initial condition x = -(y + z).
Our next step involves simplifying the expression for p, which is given by p = x^5 / (x^3 * y * z). Simplification not only makes the expression more manageable but also reveals underlying relationships between the variables. Here's how we can simplify p:
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Apply the Quotient Rule for Exponents: We can simplify the expression by dividing x^5 by x^3. According to the quotient rule for exponents, when dividing like bases, we subtract the exponents. Thus, x^5 / x^3 = x^(5-3) = x^2. This simplifies our expression for p to:
p = x^2 / (y * z)
This simplification significantly reduces the complexity of the expression. Now, we have p expressed in terms of x^2, y, and z, which makes it easier to work with in subsequent steps. We will now move on to simplifying the expression for q in a similar fashion.
Now, let's focus on simplifying the expression for q, which is given by q = y^5 / (x * y^3 * z). Similar to the simplification of p, we will use exponent rules and algebraic manipulation to reduce this expression to its simplest form. The steps are as follows:
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Apply the Quotient Rule for Exponents: We can simplify the expression by dividing y^5 by y^3. Using the quotient rule for exponents, where we subtract the exponents when dividing like bases, we get y^5 / y^3 = y^(5-3) = y^2. This simplifies the expression for q to:
q = y^2 / (x * z)
With this simplification, the expression for q is now much cleaner and easier to handle. We have expressed q in terms of y^2, x, and z. This simplified form will be crucial when we later combine p, q, and r. Next, we will simplify the expression for r following a similar process.
Moving on to the expression for r, which is defined as r = z^2 / (x * y * z^3), our goal is to simplify it using similar algebraic techniques as before. This will help us in the ultimate calculation of a^p * a^q * a^r. Here are the simplification steps:
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Apply the Quotient Rule for Exponents: We simplify the expression by dividing z^2 by z^3. Applying the quotient rule for exponents, we subtract the exponents, giving us z^2 / z^3 = z^(2-3) = z^(-1). This can also be written as 1/z. Thus, the expression for r simplifies to:
r = 1 / (x * y * z)
By simplifying, we now have r expressed in a much more manageable form, involving the reciprocal of the product of x, y, and z. This simplification completes the individual simplification of p, q, and r. We are now well-prepared to combine these simplified expressions and use the given condition x = -(y + z) to further simplify the problem.
Now that we have simplified p, q, and r, the next crucial step is to calculate the sum p + q + r. This is a critical step because the sum will allow us to simplify the expression a^p * a^q * a^r using exponent rules. Recall the simplified forms:
- p = x^2 / (y * z)
- q = y^2 / (x * z)
- r = 1 / (x * y)
Let's add these expressions together:
p + q + r = (x^2 / (y * z)) + (y^2 / (x * z)) + (1 / (x * y))
To add these fractions, we need to find a common denominator. The least common denominator (LCD) for these fractions is x * y * z. Now, we rewrite each fraction with the common denominator:
p + q + r = (x^3 / (x * y * z)) + (y^3 / (x * y * z)) + (z / (x * y * z))
Now that the fractions have a common denominator, we can add the numerators:
p + q + r = (x^3 + y^3 + z) / (x * y * z)
Now, we need to use the given condition x = -(y + z) to further simplify this expression. This substitution is key to unlocking the final solution.
We have reached a pivotal point in our solution where we will use the given condition x = -(y + z) to simplify the expression for p + q + r. This substitution is crucial for relating the variables and simplifying the overall expression. Our expression for p + q + r is:
p + q + r = (x^3 + y^3 + z) / (x * y * z)
Substitute x with -(y + z):
p + q + r = ((-(y + z))^2 / (y * z)) + (y^2 / (-(y + z) * z)) + (z^2 / (-(y + z) * y))
This substitution leads to:
p + q + r = ((-(y + z))^3 + y^3 + z^3) / (-(y + z) * y * z)
Let's expand (-(y + z))^3:
(-(y + z))^3 = -(y^3 + 3y^2z + 3yz^2 + z^3)
Substitute this back into the expression:
p + q + r = (-(y^3 + 3y^2z + 3yz^2 + z^3) + y^3 + z^3) / (-(y + z) * y * z)
Simplify the numerator:
p + q + r = (-y^3 - 3y^2z - 3yz^2 - z^3 + y^3 + z^3) / (-(y + z) * y * z)
p + q + r = (-3y^2z - 3yz^2) / (-(y + z) * y * z)
Factor out -3yz from the numerator:
p + q + r = -3yz(y + z) / (-(y + z) * y * z)
Now, cancel out the common terms -(y + z), y, and z:
p + q + r = (-3yz(y + z)) / (-yz(y + z))
p + q + r = 3
Therefore, the sum p + q + r simplifies to 3. This significant result now allows us to easily compute the final answer.
With the sum p + q + r calculated to be 3, we are now in a position to find the value of a^p * a^q * a^r. This calculation will make use of the fundamental rules of exponents, which state that when multiplying exponential expressions with the same base, we add the exponents. We have:
a^p * a^q * a^r = a^(p + q + r)
We have already determined that p + q + r = 3. Therefore, we substitute this value into the expression:
a^(p + q + r) = a^3
Thus, the value of a^p * a^q * a^r is a^3. This completes the solution to our problem.
In conclusion, by systematically simplifying the expressions for p, q, and r, and by utilizing the given condition x = -(y + z), we were able to determine that the value of a^p * a^q * a^r is a^3. This problem underscores the importance of algebraic manipulation, the application of exponent rules, and strategic substitution in solving complex mathematical problems. The step-by-step approach we followed ensures clarity and provides a robust method for tackling similar problems. This detailed solution not only answers the specific question but also enhances our understanding of mathematical problem-solving techniques.