Problem 36 - Algebra and Trigonometry

Question

Solve the equation both algebraically and graphically. $$ 6(x+2)^{5}=64 $$

Step-by-Step Solution

Answer

The solution is \(x = \sqrt[5]{\frac{32}{3}} - 2\).

Step 1: Isolate the Power

To solve the equation algebraically, start by dividing both sides of the equation by 6 to isolate the term with the power: \[6(x+2)^{5}=64\] Divide by 6: \[(x+2)^{5} = \frac{64}{6} = \frac{32}{3}\]

Step 2: Take the Fifth Root

Take the fifth root of both sides to eliminate the exponent: \[(x+2) = \sqrt[5]{\frac{32}{3}}\] Note: You need to calculate the fifth root of \(\frac{32}{3}\).

Step 3: Solve for x

Subtract 2 from both sides to solve for \(x\): \[x = \sqrt[5]{\frac{32}{3}} - 2\] Calculate the fifth root and simplify the expression to find the exact value of \(x\).

Step 4: Graph the Function and Verify

To solve graphically, plot the function \(y = 6(x+2)^5\) and a horizontal line \(y = 64\) on the same set of axes. Look for the x-coordinate of the point where the two graphs intersect; this will verify the algebraic solution.

Key Concepts

Solving Equations

Solving exponential equations can seem daunting at first, but breaking the process into steps makes it manageable. Let's start by understanding the equation given: 6(x+2)^5 = 64.The goal here is to solve for the variable \(x\). When dealing with exponential equations, it's crucial to isolate the exponential expression first. In this case, the term \((x+2)^5\) is where the power lies.To isolate it, you need to eliminate any coefficients or constants. You can achieve this by dividing both sides of the equation by 6, resulting in:(x+2)^5 = \frac{32}{3}.Once isolated, the next step is to remove the exponent by taking the fifth root on both sides. This helps you "release" the \(x+2\) from the fifth power, giving:x+2 = \sqrt[5]{\frac{32}{3}}.Finally, to solve for \(x\), subtract 2 from both sides:x = \sqrt[5]{\frac{32}{3}} - 2.This algebraic method efficiently breaks down the equation into a series of simpler steps, ultimately revealing the solution for \(x\).

Graphical Solutions

Graphical solutions provide a visual representation of the problem, which can dramatically help in understanding it. To solve the equation \(6(x+2)^5 = 64\) graphically, you would plot the following functions on a graph:

• y = 6(x+2)^5
• y = 64

By drawing these graphs on the same set of axes, you can visually inspect the intersection point. The x-coordinate of the intersection gives the solution. This intersection represents the value of \(x\) where both functions yield the same output. Graphical methods can be especially useful when dealing with complex equations where algebraic methods become cumbersome or impractical. They offer a way to verify solutions derived by other methods without complex calculations. With tools like graphing calculators or software, you can quickly plot and identify the intersection point, aiding faster and sometimes more intuitive problem-solving.

Algebraic Methods

Algebraic methods are powerful tools for solving equations analytically. They involve manipulating the equation through a set of standard procedures to isolate the variable of interest.For exponential equations like \(6(x+2)^5 = 64\), these methods entail reducing the equation step by step:

• First, eliminate coefficients that multiply the power, here by dividing by 6.
• Then, take the root corresponding to the exponent, which in this case is the fifth root.
• Finally, subtract any constants to solve for the variable.

These steps involve key algebraic principles like performing inverse operations, simplifying expressions, and transforming equations. Each stage simplifies the problem, rendering the solution more apparent as you progress.Using algebraic methods not only sharpens problem-solving skills but also builds a deeper understanding of the mathematical relationships within equations. They form the backbone of algebra, empowering students to tackle a wide variety of problems, from simple linear equations to more complex exponential ones.