Problem 66 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 4(x+2)^{5}=1 $$

Step-by-Step Solution

Answer

The real solution is approximately \(x \approx -1.2414\).

Step 1: Isolate the Exponential Expression

Start by dividing both sides of the equation by 4 to isolate the exponential expression. This gives us: \[(x+2)^5 = \frac{1}{4}\]

Step 2: Apply the Fifth Root

Take the fifth root of both sides of the equation to eliminate the power of 5, yielding: \[x+2 = \sqrt[5]{\frac{1}{4}}\]

Step 3: Solve for x

Subtract 2 from both sides to solve for \(x\). Hence: \[x = \sqrt[5]{\frac{1}{4}} - 2\]

Step 4: Simplify the Expression

Compute or simplify \(\sqrt[5]{\frac{1}{4}}\) if possible. The fifth root of \(\frac{1}{4}\) is a real number approximated as: \[x \approx 0.7586 - 2\]Thus, \[x \approx -1.2414\]

Key Concepts

Understanding Exponents and Power

Exponents are a way of expressing repeated multiplication. When you see an expression like \((x+2)^5\), it means \((x+2)\) is multiplied by itself five times.
In mathematics, exponents are important because they help simplify expressions. For instance, instead of writing \((x+2) \times (x+2) \times (x+2) \times (x+2) \times (x+2)\), we use \((x+2)^5\).
This not only makes the expression shorter but also easier to read.

• The base is the number or variable being multiplied, which is \((x+2)\) here.
• The exponent is the small number above the base, indicating how many times the base is multiplied by itself. In this exercise, the exponent is 5.

To work effectively with exponents, you need to understand laws like the power of a power, product of powers, and so on.

Finding Real Solutions to Equations

A real solution of an equation is any solution that is a real number. Real numbers include all the numbers you usually work with, such as integers, fractions, and decimals. There are also numbers like \(-1.2414\), which can be found in our example when solving the given equation.

To find real solutions, you often isolate the variable and simplify as much as possible. In this case, we divided both sides by 4 to isolate the portion with the variable \((x+2)^5\). Then, we went on to reduce the complexity of the problem by applying the fifth root.

• The goal is to get \(x\) alone on one side of the equation.
• Once \(x\) is isolated, if it represents a real number, you've found your real solution.

The solutions are explicitly real when they don’t involve imaginary numbers, which occur when dealing with negative numbers under even roots.

The Role of the Fifth Root in Solving Equations

Taking the fifth root is a method used to solve equations that involve a number raised to the fifth power. The fifth root is the number that, when raised to the power of five, gives the original number.

This concept is similar to square roots but involves five as the power instead of two.

• If you have \((x+2)^5 = \frac{1}{4}\), taking the fifth root of both sides simplifies it to \(x+2 = \sqrt[5]{\frac{1}{4}}\).
• The fifth root of \(\frac{1}{4}\) is approximately \(0.7586\), a real number.

Once you find \(\sqrt[5]{\frac{1}{4}}\), subtracting 2 will give you the solution for \(x\), which is approximately
\(-1.2414\). Using the fifth root is essential here because it neutralizes the power of 5, making it easy to solve for the unknown variable \(x\).