Problem 10 - Algebra 2

Question

Solve. \(3(x+3)^{\frac{3}{4}}=81\)

Step-by-Step Solution

Answer

The solution to the equation \(3(x+3)^{\frac{3}{4}}=81\) is \(x = 27^{\frac{4}{3}} - 3\)

Step 1: Eliminate the coefficient from the left side

Start by dividing both sides of the equation by 3 to isolate the expression on the left side: \((x+3)^{\frac{3}{4}} = 27\)

Step 2: Get rid of the cube root from equation

Since \((x+3)^{\frac{3}{4}}\) is equivalent to \(\sqrt[4]{(x+3)^3}\), we can eliminate the cube root by raising both sides of the equation to the 4/3 power: \((x+3) = 27^{\frac{4}{3}}\)

Step 3: Solve for x

Now, we can solve for x by subtracting 3 from both sides of the equation: \(x = 27^{\frac{4}{3}} - 3\)

Key Concepts

Rational Exponents

Rational exponents offer a different way to represent roots and powers using fractions. When you see an exponent like \( \frac{3}{4} \), it signifies a combination of both a power and a root, which can initially seem tricky but is quite useful for simplifying certain expressions.
To understand a rational exponent, split it into two parts:

• The numerator 3 is the power applied to the base.
• The denominator 4 is the root taken of the result.

In the expression \((x+3)^{\frac{3}{4}}\), you are essentially taking the fourth root (because of the denominator) of \((x+3)^3\).
Rational exponents make expressing roots and powers easier and more efficient. Instead of writing \( \sqrt[4]{(x+3)^3} \), we write \((x+3)^{\frac{3}{4}}\). This form is particularly handy in calculus and higher mathematics.

Equation Simplification

Equation simplification is the process of making an equation easier to solve or understand. When working with radical expressions and rational exponents, this might involve removing coefficients or using algebraic properties to isolate terms.
In the given problem, the first step is to simplify the expression by removing the coefficient 3. This is done by dividing both sides of the equation by 3 to get:
\((x+3)^{\frac{3}{4}} = 27\).
Simplifying equations often involves:

• Eliminating fractions or decimals.
• Combining like terms or reducing coefficients.

These simplifications make the equation more straightforward, allowing us to focus on the variable we need to solve for. Simplification is crucial as it often reveals a clearer path to the solution.

Exponentiation

Exponentiation is the mathematical operation involving repeated multiplication of a number by itself. It's essential when working with powers and roots, as shown in the exercise.
When dealing with the equation \( (x+3) = 27^{\frac{4}{3}} \), you exponentiate 27 by the fractional power \( \frac{4}{3} \). This tells us to:

• Take the cube root of 27 first (the denominator 3), which results in 3.
• Raise this result (3) to the power of 4 (the numerator), giving us 81.

Exponentiation, especially with rational exponents, helps simplify roots and powers in one operation. Understanding this concept allows you to solve equations effectively by breaking down complex powers into sequential, simpler steps.