Problem 60 - Algebra 2

Question

Solve \((x+2)^{\frac{3}{4}}=27\)

Step-by-Step Solution

Answer

The solution to the equation \((x+2)^{\frac{3}{4}}=27\) is \(x = 79\).

Step 1: Isolate the term with the variable

The first step is to isolate the term with the variable. We already have \((x+2)^{\frac{3}{4}}=27\) and thus the term with the variable \(x\) is isolated.

Step 2: Remove the radical

Next, raise both sides of the equation to the power that will remove the radical from the term with the variable. The term \((x+2)^{\frac{3}{4}}\) is equivalent to the \(4th\) root of \((x+2)^3\). If we raise both sides to the power of \(\frac{4}{3}\), we will be able to remove the radical. Doing so gives \(((x+2)^{\frac{3}{4}})^{\frac{4}{3}}=(27)^{\frac{4}{3}}\). This reduces to \(x+2 = 27^{\frac{4}{3}}\).

Step 3: Calculate the constant

Calculate the value of \(27^{\frac{4}{3}}\). Using the properties of exponents, this evaluates to \(3^4 = 81\). So substitution results in \(x+2 = 81\).

Step 4: Solve for the variable

Finally, solve for the variable by subtracting \(2\) from both sides of the equation. This gives \(x = 81 - 2 = 79\).

Key Concepts

Radicals

Radicals are expressions that involve root operations, such as square roots or cube roots. When we encounter a term like \((x+2)^{\frac{3}{4}}\), it suggests a combination of exponentiation and a radical. Specifically, the term indicates that we take the cube root of \((x+2)\) raised to the 3rd power and then consider its fourth root.
In the context of solving exponential equations, the key idea is to eliminate radicals by applying inverse operations.

• Understand the root: The exponent \(\frac{3}{4}\) indicates operations involving both the cube (3) and root (4).
• Apply inverse operations: To eliminate the fourth root, we need to raise both sides to a power that cancels out the fraction, such as \(\frac{4}{3}\) in this case.
• Simplify the process: By manipulating the expression, we transform a radical into simpler components that are easier to handle algebraically.

These steps can be crucial for solving the equation and finding the correct value of the variable. By eradicating the complexity of radicals, solutions become more straightforward and accessible.

Exponents

Exponents denote repeated multiplication and play a vital role in equations where terms are raised to a certain power. In our exercise, the term \((x+2)^{\frac{3}{4}}\) involves an exponent that is a fractional value. This fraction indicates both exponentiation and rooting (as explained earlier).
Understanding the behavior of exponents is essential for manipulating and solving equations.

• Fractional exponents: Express roots as exponents; for example, \(a^{\frac{m}{n}}\) represents the nth root of \(a\) raised to the mth power.
• Properties of exponents: Utilize properties like \((a^m)^n = a^{m \cdot n}\) to simplify expressions and solve for variables.
• Calculation: In this case, transforming the exponent from \(\frac{3}{4}\) to \(\frac{4}{3}\) aids in simplifying the equation by removing the root aspect.

Accurate manipulation of such expressions is crucial. By understanding how exponents work, one can effectively solve problems involving exponential equations.

Isolating Variables

Isolating variables is a fundamental process in solving equations, referred to as 'solving for \(x\)'. The goal is to get the variable alone on one side of the equation. In the given solution, this process is straightforward because \((x+2)\) is already isolated upon starting.Here are key points for isolating variables:

• Recognizing isolation: The initial step involves ensuring the term with the variable is by itself, which means rearranging the equation if necessary.
• Step-by-step simplification: Once the radical or exponent is dealt with, further simplifying the equation by performing operations like addition or subtraction isolates the variable completely.
• Final steps: After the exponential manipulation, we found \(x + 2 = 81\). To isolate \(x\), we subtract 2 from both sides, resulting in \(x = 79\).

Ensuring variables are isolated correctly is a straightforward yet critical step in solving any equation. It allows us to find the solution directly and verify it quickly once other complexities are resolved.