Solve For Positive Solution: $5x^{9/8} + 11 = 671088651$

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In the realm of mathematics, solving equations is akin to unlocking puzzles, revealing underlying truths and relationships. Today, we embark on a journey to find the positive solution to a specific algebraic equation: 5 x^{rac{9}{8}}+11=671088651. This equation, while seemingly complex, follows a systematic approach to unravel its mysteries. Our goal is to isolate the variable $x$ and determine its positive value that satisfies this equality. Let's begin by understanding the structure of the equation and the operations we'll employ to reach our solution. We are looking for a number $x$ such that when raised to the power of rac{9}{8} and then multiplied by 5, and finally added to 11, results in the large number 671088651. This process involves inverse operations, a fundamental concept in algebra, where we undo each operation performed on $x$ to reveal its original value.

Our first step in solving the equation 5 x^{rac{9}{8}}+11=671088651 is to isolate the term containing $x$. Currently, the term 5 x^{rac{9}{8}} is being added to 11. To undo this addition, we perform the inverse operation: subtraction. We will subtract 11 from both sides of the equation to maintain the balance. This gives us:

5 x^{rac{9}{8}} = 671088651 - 11

5 x^{rac{9}{8}} = 671088640

Now, the term with $x$ is being multiplied by 5. To isolate x^{rac{9}{8}}, we need to undo this multiplication by performing the inverse operation: division. We will divide both sides of the equation by 5:

x^{rac{9}{8}} = rac{671088640}{5}

x^{rac{9}{8}} = 134217728

At this stage, we have $x$ raised to the power of rac{9}{8}. To solve for $x$, we need to eliminate this fractional exponent. This involves raising both sides of the equation to the reciprocal power of rac{9}{8}, which is rac{8}{9}. The reciprocal power is crucial because when we raise a power to another power, we multiply the exponents. So, (rac{9}{8}) imes (rac{8}{9}) = 1, effectively leaving us with $x^1$, which is simply $x$.

Therefore, we raise both sides of the equation x^{rac{9}{8}} = 134217728 to the power of rac{8}{9}:

(x^{rac{9}{8}})^{rac{8}{9}} = (134217728)^{rac{8}{9}}

x = (134217728)^{rac{8}{9}}

To calculate (134217728)^{rac{8}{9}}, we can think of this as taking the 9th root of 134217728 and then raising the result to the 8th power. Alternatively, we can recognize that 134217728 is a power of 2. Let's find which power it is. We can do this by repeatedly dividing by 2 or by using logarithms. It turns out that $134217728 = 2^{27}$.

So, the equation becomes:

x = (2^{27})^{rac{8}{9}}

Using the rule of exponents $(a^m)^n = a^{m imes n}$, we multiply the exponents:

x = 2^{27 imes rac{8}{9}}

x = 2^{rac{27 imes 8}{9}}

x = 2^{rac{216}{9}}

$x = 2^{24}$

Now, we need to calculate $2^{24}$. This is a large number, but manageable with a calculator or by breaking it down. $2^{10} = 1024$. We can express $2^{24}$ as $2^{10} imes 2^{10} imes 2^4$ or $2^{20} imes 2^4$.

$2^{24} = (2^{10})^2 imes 2^4$

$2^{24} = (1024)^2 imes 16$

$(1024)^2 imes 16 = 1048576 imes 16$

$1048576 imes 16 = 16777216$

So, the positive solution for $x$ is 16,777,216.

To verify our solution, we can substitute $x = 16,777,216$ back into the original equation 5 x^{rac{9}{8}}+11=671088651. Since $x = 2^{24}$, we have:

x^{rac{9}{8}} = (2^{24})^{rac{9}{8}} = 2^{24 imes rac{9}{8}} = 2^{rac{216}{8}} = 2^{27}

We already established that $2^{27} = 134217728$. Now, let's plug this back into the original equation:

$5 imes (134217728) + 11$

$671088640 + 11 = 671088651$

This matches the right side of the original equation, confirming that our positive solution is indeed correct. The journey through solving this equation highlights the power of inverse operations and exponent rules in simplifying complex mathematical expressions. Understanding these principles allows us to tackle a wide array of algebraic challenges with confidence and precision.

Understanding Exponents and Roots

The core of solving this equation lies in understanding fractional exponents and their relationship with roots. A fractional exponent, like rac{9}{8} in x^{rac{9}{8}}, signifies a combination of a root and a power. Specifically, x^{rac{m}{n}} can be interpreted as $( ext{the } n ext{-th root of } x)^m$ or $ ext{the } n ext{-th root of } (x^m)$. In our case, x^{rac{9}{8}} means $( ext{the 8th root of } x)^9$ or $ ext{the 8th root of } (x^9)$. For solving, it's generally easier to work with the root first, if possible, as it reduces the magnitude of the numbers involved.

The concept of a reciprocal power, as used when we raised both sides to the rac{8}{9} power, is a critical tool for isolating variables with fractional exponents. When you raise a power to another power, you multiply the exponents. If the exponent is rac{a}{b}, its reciprocal is rac{b}{a}. Multiplying these together, rac{a}{b} imes rac{b}{a} = 1. This is precisely what we need to get the variable $x$ by itself. In our problem, we had x^{rac{9}{8}} and we raised it to the power of rac{8}{9}. The resulting exponent for $x$ became (rac{9}{8}) imes (rac{8}{9}) = 1, leaving us with just $x$.

The Significance of the Positive Solution

The question specifically asked for the positive solution. This is important because some equations, particularly those involving even roots or certain powers, can have both positive and negative solutions. For instance, if we had $x^2 = 4$, the solutions would be $x=2$ and $x=-2$. However, when dealing with fractional exponents like x^{rac{9}{8}}, the operation is typically defined to yield a single, principal root. In this context, x^{rac{9}{8}} implies the principal (positive) 8th root of $x$, raised to the 9th power. Consequently, when we solve for $x$, we naturally arrive at a positive value, provided the right-hand side of the equation after simplification is positive.

If the equation had been structured differently, for example, if it involved an even root explicitly like (rac{1}{8}) power where $x$ could be negative, we might have had to consider the domain of the function and whether negative values are permissible or yield valid results. However, in x^{rac{9}{8}}, the presence of the 8th root means $x$ must be non-negative for the expression to be defined in the real number system. Therefore, seeking a positive solution is consistent with the nature of the expression.

Powers of Two and Their Importance

Recognizing that $134217728$ is a power of two ($2^{27}$) was a significant shortcut in solving (134217728)^{rac{8}{9}}. Powers of two are fundamental in computer science and many areas of mathematics, and they often appear in problems involving exponents. If we hadn't recognized this, we would have had to calculate the 9th root of $134217728$ and then raise it to the 8th power. This would involve using a calculator and potentially dealing with rounding errors if the root wasn't a clean integer. For example, the 9th root of $134217728$ is exactly 2. So, (134217728)^{rac{1}{9}} = 2. Then, $2^8 = 256$. Wait, there was a miscalculation in my thought process. Let me re-evaluate.

Ah, the step was x = (134217728)^{rac{8}{9}}. My previous calculation was $x = 2^{24}$. Let's re-trace the exponent multiplication: 27 imes rac{8}{9}. This indeed results in $24$. So $x=2^{24}$.

Let's recalculate $2^{24}$ carefully. $2^{10} = 1024$. $2^{20} = (2^{10})^2 = 1024^2 = 1048576$. Then $2^{24} = 2^{20} imes 2^4 = 1048576 imes 16$.

$1048576 imes 10 = 10485760$

$1048576 imes 6 = 6291456$

$10485760 + 6291456 = 16777216$. Yes, this calculation is correct. The value $16,777,216$ is indeed $2^{24}$.

My previous verification step also used $2^{27}$ correctly. It seems I had a momentary lapse in the intermediate calculation check. The method was sound, and the final result holds. The ease of calculation, however, heavily relies on identifying $134217728$ as $2^{27}$. This is a common technique in mathematical problem-solving: look for recognizable patterns or bases, especially powers of small integers like 2, 3, or 10.

Conclusion

We have successfully navigated the steps to solve the equation 5 x^{rac{9}{8}}+11=671088651 and found its positive solution to be $x = 16,777,216$. This involved the fundamental algebraic principles of isolating the variable through inverse operations: subtracting 11, dividing by 5, and then applying the reciprocal of the fractional exponent. The recognition of $134217728$ as $2^{27}$ greatly simplified the final calculation of (134217728)^{rac{8}{9}}, leading to $2^{24}$. This problem serves as an excellent illustration of how combining knowledge of algebraic manipulation, exponent rules, and number properties can lead to the elegant solution of complex-looking equations. The mathematical journey from a complex equation to a clear numerical answer is a testament to the logical and structured nature of mathematics.

For those interested in delving deeper into the fascinating world of algebra and number theory, exploring resources like Brilliant.org can provide further insights and practice problems. They offer interactive lessons and explanations that can enhance your understanding of these concepts.