Solving Equations With Rational Exponents A Detailed Guide

Understanding Rational Exponents

Before diving into solving the equation $(x-3)^{2/3} = 16$, it's crucial to understand what rational exponents mean. A rational exponent is an exponent that can be expressed as a fraction, such as $m/n$, where m and n are integers. In general, $a^{m/n}$ can be interpreted in two equivalent ways:

1. $(a^{1/n})^m$, which means taking the n-th root of a and then raising the result to the power of m.
2. $(a^m)^{1/n}$, which means raising a to the power of m and then taking the n-th root of the result.

Both interpretations are valid and will yield the same result, but one might be easier to compute than the other depending on the specific numbers involved. For example, if we have $8^{2/3}$, we can either calculate the cube root of 8 first (which is 2) and then square it ($2^2 = 4$), or we can square 8 first ($8^2 = 64$) and then find the cube root of 64 (which is 4). In this case, taking the cube root first simplifies the calculation.

Understanding this concept is fundamental to solving equations with rational exponents. The ability to switch between the radical form and the exponential form allows us to manipulate equations more easily and apply appropriate algebraic techniques. In the given equation, $(x-3)^{2/3} = 16$, we will use this understanding to isolate x and find the solutions. Rational exponents are not just a mathematical curiosity; they appear in various fields, including physics, engineering, and computer science, making it essential to grasp their meaning and application. Therefore, a solid understanding of rational exponents lays the groundwork for more advanced mathematical concepts and problem-solving skills. The manipulation of these exponents often involves the use of exponent rules, such as the power of a power rule, which states that $(a^m)^n = a^{mn}$. These rules, when applied correctly, help simplify complex expressions and equations. In the context of our equation, we will leverage these principles to undo the rational exponent and isolate the variable x. By doing so, we can transform the equation into a more manageable form, enabling us to determine the possible values of x that satisfy the original equation.

Solving the Equation $(x-3)^{2/3} = 16$

Now, let's tackle the equation $(x-3)^{2/3} = 16$ step by step. Our goal is to isolate x, but we first need to deal with the rational exponent of 2/3. To do this, we'll use the property that raising a power to its reciprocal will cancel out the exponent. The reciprocal of 2/3 is 3/2, so we will raise both sides of the equation to the power of 3/2.

This gives us:

$((x-3)^{2/3})^{3/2} = 16^{3/2}$

On the left side, the exponents (2/3) and (3/2) multiply to 1, effectively canceling each other out, leaving us with:

$x - 3 = 16^{3/2}$

Now, we need to evaluate $16^{3/2}$. Recall that $16^{3/2}$ can be interpreted as $(16^{1/2})^3$ or $(16^3)^{1/2}$. It's generally easier to take the root first, so we'll go with $(16^{1/2})^3$. The square root of 16 is 4, so we have:

$16^{3/2} = (16^{1/2})^3 = 4^3 = 64$

However, we must also consider the negative root. When dealing with rational exponents where the denominator is even, we need to account for both positive and negative roots. So, $16^{1/2}$ can be either 4 or -4. Therefore, we also need to consider $(-4)^3$, which is -64.

So, we have two possibilities:

$x - 3 = 64$ and $x - 3 = -64$

Solving for x in each case:

For $x - 3 = 64$, we add 3 to both sides to get $x = 67$.

For $x - 3 = -64$, we add 3 to both sides to get $x = -61$.

Thus, we have two potential solutions: $x = 67$ and $x = -61$. It is essential to check these solutions in the original equation to ensure they are valid because raising to a rational power can sometimes introduce extraneous solutions. This is due to the fact that the process of raising both sides of an equation to a power can sometimes create solutions that do not satisfy the original equation. This is particularly true when dealing with even roots, as negative numbers raised to an even power become positive, potentially leading to discrepancies. Therefore, the final step in solving equations with rational exponents should always involve substituting the obtained solutions back into the original equation to confirm their validity. This practice ensures the accuracy of the solution set and prevents the inclusion of values that are mathematically inconsistent with the initial equation.

Checking the Solutions

It's crucial to check our solutions in the original equation $(x-3)^{2/3} = 16$ to ensure they are valid. Let's check $x = 67$ first:

$(67 - 3)^{2/3} = (64)^{2/3}$

We can rewrite $64^{2/3}$ as $(64^{1/3})^2$. The cube root of 64 is 4, so we have:

$(4)^2 = 16$

So, $x = 67$ is a valid solution.

Now, let's check $x = -61$:

$(-61 - 3)^{2/3} = (-64)^{2/3}$

We can rewrite $(-64)^{2/3}$ as $((-64)^{1/3})^2$. The cube root of -64 is -4, so we have:

$(-4)^2 = 16$

So, $x = -61$ is also a valid solution. Therefore, the process of verifying solutions is not just a formality but a necessary step to guarantee the correctness of the answer. It reinforces the understanding of the original problem and ensures that the algebraic manipulations have not introduced any inconsistencies. In addition to identifying extraneous solutions, the verification step also provides an opportunity to review the solution process and reinforce the mathematical concepts involved. This practice is particularly beneficial in complex problems where multiple steps and operations are required to arrive at the solution. By systematically checking each solution, mathematicians and problem-solvers can build confidence in their results and avoid common errors. Moreover, the act of verification promotes a deeper understanding of the relationship between the equation and its solutions, fostering a more robust grasp of the underlying mathematical principles. This rigorous approach is essential not only in academic settings but also in real-world applications where accuracy and reliability are paramount.

Final Answer

Both $x = 67$ and $x = -61$ satisfy the original equation. Therefore, the solution set is {-61, 67}.

So, the correct choice is:

A. The solution set is {-61, 67}.