Solving The Equation 2(x+7)^(2/3) = 8 A Step-by-Step Guide

Algebraic equations are fundamental to mathematics and various fields of science and engineering. Mastering the techniques to solve these equations is crucial for anyone pursuing studies or careers in these areas. This article provides a step-by-step guide on how to solve the equation $2(x+7)^{\frac{2}{3}}=8$, along with a detailed explanation to ensure clarity and understanding. Solving equations like this involves understanding fractional exponents, isolating variables, and applying algebraic manipulations to arrive at the correct solution. In this comprehensive guide, we will break down each step, providing insights and strategies for tackling similar problems. Whether you're a student preparing for an exam or someone looking to refresh your math skills, this article aims to equip you with the knowledge and confidence to solve complex algebraic equations effectively.

The initial step in solving the equation $2(x+7)^{\frac{2}{3}}=8$ involves isolating the term with the fractional exponent. This is a critical step in simplifying the equation and making it easier to work with. To achieve this isolation, we need to eliminate the coefficient that is multiplying the exponential term. In this case, the coefficient is 2. The isolating the exponential term is essential because it sets the stage for the subsequent steps, which involve dealing with the fractional exponent itself. By dividing both sides of the equation by 2, we maintain the balance of the equation while moving closer to isolating the term containing the variable. This approach is a cornerstone of algebraic manipulation, ensuring that we can apply further operations without altering the equation's fundamental properties. This preliminary step simplifies the equation, making it more manageable and paving the way for the next phase of the solution. Once the exponential term is isolated, we can then focus on addressing the fractional exponent and ultimately solve for the variable x. The process of isolating terms is a common strategy in solving various types of equations, including those involving radicals, logarithms, and trigonometric functions. Therefore, mastering this technique is crucial for anyone seeking proficiency in algebra and related mathematical disciplines. By carefully executing each step, we move closer to the solution and develop a deeper understanding of the underlying mathematical principles.

Let's dive into the step-by-step solution of the equation. We will start by dividing both sides by 2 to isolate the exponential term, then address the fractional exponent by raising both sides to the reciprocal power, and finally solve for x. This methodical approach will help you understand the underlying principles and apply them to other algebraic equations. Each step is crucial and builds upon the previous one, demonstrating the logical progression required for solving complex problems. Understanding these steps not only helps in finding the solution but also enhances your problem-solving skills in mathematics. By breaking down the problem into manageable parts, we make the process less daunting and more accessible. This approach is particularly useful for students who are learning algebra, as it provides a clear framework for tackling similar equations. The following detailed steps will guide you through the solution, ensuring you grasp every aspect of the process.

Step 1: Isolate the Exponential Term

To isolate the exponential term $(x+7)^{\frac{2}{3}}$, we need to eliminate the coefficient 2. This is achieved by dividing both sides of the equation by 2. This maintains the equality and simplifies the equation, making it easier to work with. The act of dividing both sides by the same number is a fundamental algebraic principle, ensuring that the equation remains balanced. When we isolate the exponential term, we pave the way for dealing with the fractional exponent, which is the next critical step in solving the equation. This process is analogous to peeling away the layers of an onion, each step revealing a simpler form of the equation. Isolating terms is a common strategy in solving various types of equations, including linear, quadratic, and exponential equations. By mastering this technique, you can approach a wide range of mathematical problems with confidence. The initial equation, $2(x+7)^{\frac{2}{3}}=8$, transforms into a simpler form, setting the stage for further algebraic manipulations. This step is crucial in the overall solution process, as it allows us to focus on the term containing the variable x. The resulting equation is more manageable and allows us to proceed with the next logical step in the solution.

Detailed Calculation for Step 1

Starting with the original equation, $2(x+7)^{\frac{2}{3}}=8$, we divide both sides by 2:

$\frac{2(x+7)^{\frac{2}{3}}}{2} = \frac{8}{2}$

This simplifies to:

$(x+7)^{\frac{2}{3}} = 4$

This new form of the equation is significantly easier to work with. By isolating the exponential term, we've effectively reduced the complexity of the problem. The next step will involve dealing with the fractional exponent, which will ultimately lead us to the solution for x. This meticulous approach of isolating terms and simplifying the equation is a key technique in algebra. The detailed calculation above demonstrates the clarity and precision required in mathematical problem-solving. By carefully executing each step, we minimize the risk of errors and build a solid foundation for further calculations. The result of this step is a critical milestone in the solution process, bringing us closer to the final answer. This process underscores the importance of understanding fundamental algebraic principles and applying them systematically to solve equations. The next step will build upon this result, leading us toward the solution for the variable x.

Step 2: Eliminate the Fractional Exponent

Now that we have $(x+7)^{\frac{2}{3}} = 4$, we need to eliminate the fractional exponent. To do this, we raise both sides of the equation to the reciprocal of the exponent, which is $\frac{3}{2}$. This is a crucial step in solving equations with fractional exponents. Raising a power to its reciprocal effectively cancels out the exponent, allowing us to isolate the base. The principle behind this step lies in the properties of exponents: $(a^m)^n = a^{m*n}$. By raising the term with the fractional exponent to its reciprocal power, we transform the exponent to 1, thus eliminating it. This technique is widely used in solving equations involving radicals and fractional powers. The reciprocal exponent effectively "undoes" the original exponent, making the equation more straightforward to solve. This step not only simplifies the equation but also demonstrates a fundamental concept in algebra. By understanding how to manipulate exponents, you can solve a wide variety of mathematical problems. The process of raising both sides to a power maintains the equality of the equation, ensuring that our solution remains valid. The following detailed calculation will illustrate how this step transforms the equation and brings us closer to finding the value of x.

Detailed Calculation for Step 2

Starting with $(x+7)^{\frac{2}{3}} = 4$, we raise both sides to the power of $\frac{3}{2}$:

$((x+7)^{\frac{2}{3}})^{\frac{3}{2}} = 4^{\frac{3}{2}}$

On the left side, the exponents multiply: $\frac{2}{3} * \frac{3}{2} = 1$, so we get:

$(x+7)^1 = 4^{\frac{3}{2}}$

The right side can be rewritten as $(4^{\frac{1}{2}})^3$. Since $4^{\frac{1}{2}} = \sqrt{4} = 2$, we have:

$4^{\frac{3}{2}} = 2^3 = 8$

Thus, the equation becomes:

$x+7 = 8$

This simplification is a significant step towards solving for x. By eliminating the fractional exponent, we have transformed the equation into a much simpler form. The detailed calculation demonstrates how the properties of exponents are applied to achieve this simplification. The process of rewriting $4^{\frac{3}{2}}$ as $2^3$ showcases a powerful technique in simplifying expressions with fractional exponents. This step not only helps in solving the equation but also enhances your understanding of exponential functions and their properties. The resulting equation, $x+7 = 8$, is a linear equation that can be easily solved for x. This transformation highlights the effectiveness of our approach in simplifying complex equations through strategic algebraic manipulations. The next step will involve isolating x to find its value.

Step 3: Solve for x

Now that we have the simplified equation $x+7 = 8$, the final step is to isolate x. To do this, we subtract 7 from both sides of the equation. This maintains the equality and gives us the value of x. Solving for x in a linear equation like this is a fundamental skill in algebra. The process of subtracting the same number from both sides ensures that the equation remains balanced and that we arrive at the correct solution. Isolating the variable is a common goal in solving various types of equations, and this step demonstrates a straightforward application of that principle. The resulting value of x will be a solution to the original equation. This step is the culmination of the previous steps, bringing us to the final answer. The process of solving for x highlights the importance of understanding basic algebraic operations and applying them correctly. By carefully executing each step, we can confidently arrive at the solution. The following calculation will show the final step in solving for x and completing the solution process.

Detailed Calculation for Step 3

Starting with $x+7 = 8$, we subtract 7 from both sides:

$x + 7 - 7 = 8 - 7$

This simplifies to:

$x = 1$

So, one solution to the equation is $x = 1$. However, we must remember that when we raised both sides of the equation to the power of $\frac{3}{2}$, we introduced the possibility of extraneous solutions because the exponent has an even number in the denominator. An extraneous solution is a solution that emerges from the process of solving the problem but is not a valid solution to the original equation. This often happens when dealing with radicals or fractional exponents. To check for extraneous solutions, we need to substitute our solution back into the original equation. The concept of extraneous solutions is crucial in algebra, particularly when dealing with equations involving radicals or fractional exponents. It highlights the importance of verifying solutions to ensure their validity. The detailed calculation above provides a clear path to finding the value of x, but the process is not complete until we verify the solution. This step demonstrates the rigor required in mathematical problem-solving. By checking for extraneous solutions, we ensure that our final answer is accurate and reliable. The next step will involve substituting the value of x back into the original equation to confirm its validity.

Step 4: Check for Extraneous Solutions

To ensure our solution is valid, we must check for extraneous solutions. This is particularly important when we have raised both sides of the equation to an even power, as this can introduce solutions that do not satisfy the original equation. Extraneous solutions arise because raising both sides of an equation to an even power can make a false statement true. For example, -2 ≠ 2, but $(-2)^2 = 2^2 = 4$. Therefore, it is essential to substitute our solutions back into the original equation to verify their validity. This process involves replacing the variable in the original equation with the solution we found and then evaluating both sides of the equation. If both sides are equal, the solution is valid; otherwise, it is extraneous. The concept of extraneous solutions is a critical aspect of solving algebraic equations, especially those involving radicals or fractional exponents. By checking for extraneous solutions, we ensure that our final answer is accurate and reliable. This step demonstrates the importance of rigor and precision in mathematical problem-solving. The following detailed calculation will show how we verify our solution by substituting it back into the original equation.

Detailed Calculation for Step 4

Substitute $x = 1$ into the original equation $2(x+7)^{\frac{2}{3}}=8$:

$2(1+7)^{\frac{2}{3}} = 8$

Simplify inside the parentheses:

$2(8)^{\frac{2}{3}} = 8$

Rewrite $8$ as $2^3$:

$2(2^3)^{\frac{2}{3}} = 8$

Multiply the exponents:

$2(2^{3*\frac{2}{3}}) = 8$

$2(2^2) = 8$

$2(4) = 8$

$8 = 8$

Since the equation holds true, $x = 1$ is a valid solution.

Now, let's consider the nature of the exponent $\frac{2}{3}$. It can be interpreted as $(\sqrt[3]{x+7})^2$. When we take the cube root, we only consider the real root. However, when we raise a number to the power of 2, we should consider both positive and negative roots. This means we should also check the case where the cube root of $(x+7)$ is -2.

So, we have:

$\sqrt[3]{x+7} = -2$

Cube both sides:

$x+7 = (-2)^3$

$x+7 = -8$

Subtract 7 from both sides:

$x = -15$

Now, we need to check if $x = -15$ is a solution.

Substitute $x = -15$ into the original equation:

$2(-15+7)^{\frac{2}{3}} = 8$

$2(-8)^{\frac{2}{3}} = 8$

Rewrite $-8$ as $(-2)^3$:

$2((-2)^3)^{\frac{2}{3}} = 8$

Multiply the exponents:

$2((-2)^{3*\frac{2}{3}}) = 8$

$2((-2)^2) = 8$

$2(4) = 8$

$8 = 8$

Since the equation holds true, $x = -15$ is also a valid solution.

The detailed calculation demonstrates the importance of checking for extraneous solutions and considering all possible roots. By substituting the solutions back into the original equation, we confirmed that both $x = 1$ and $x = -15$ are valid solutions. This thorough approach ensures that we have a complete and accurate solution to the problem. The process highlights the need for careful consideration of the properties of exponents and radicals when solving algebraic equations. The final result provides a comprehensive understanding of the solution and underscores the importance of verifying solutions in mathematical problem-solving.

In conclusion, the solutions to the equation $2(x+7)^{\frac{2}{3}}=8$ are $x = 1$ and $x = -15$. This solving equations process involved isolating the exponential term, eliminating the fractional exponent, solving for x, and checking for extraneous solutions. Each step is crucial in arriving at the correct answer. By following a systematic approach, we can tackle complex algebraic equations with confidence. The techniques demonstrated in this solution are applicable to a wide range of mathematical problems. Understanding these principles not only helps in solving equations but also enhances your overall problem-solving skills. The importance of checking for extraneous solutions cannot be overstated, as it ensures the validity of our results. This comprehensive guide provides a clear and detailed explanation of the solution process, making it accessible to students and anyone looking to improve their algebraic skills. By mastering these techniques, you can confidently approach similar problems and achieve success in mathematics.

Therefore, the correct answer is A. -15 and 1.