Problem 24 - A Graphical Approach to College Algebra

Question

Solve each equation by hand. Do not use a calculator. $$4 x^{3 / 2}+5=21$$

Step-by-Step Solution

Answer

The solution is \(x \approx 2.52\).

Step 1: Isolate the Radical Expression

The given equation is \(4x^{3/2} + 5 = 21\). Start by isolating the term with the radical expression. Subtract 5 from both sides to get \(4x^{3/2} = 16\).

Step 2: Eliminate the Coefficient of the Radical

To further isolate the radical on the left, divide both sides by 4: \(x^{3/2} = 4\). This simplifies the equation and allows us to deal with the exponent more directly.

Step 3: Eliminate the Fractional Exponent

The exponent \(\frac{3}{2}\) can be eliminated by raising both sides of the equation to the power of \(\frac{2}{3}\). Thus, \((x^{3/2})^{2/3} = 4^{2/3}\). Since \((a^m)^n = a^{m \times n}\), this simplifies to \(x^1 = x = 4^{2/3}\).

Step 4: Simplify the Right Side

Now calculate \(4^{2/3}\). First, find the cube root of 4, which is \(4^{1/3}\), and then square it. \(4^{1/3}\) can be approximated as \(\sqrt[3]{4} \approx 1.5874\). Squaring this gives us approximately \(1.5874^2 = 2.52\). So \(x \approx 2.52\).

Step 5: Calculate the Exact Value Optionally

To get a more precise value without approximation, note that exactly \(4 = 2^2\), thus \(4^{2/3} = (2^2)^{2/3} = 2^{4/3}\). Now knowing that \(2^{4/3} = (2^{1/3})^4\) and since \(2^{1/3} = \sqrt[3]{2}\), which can more accurately be simplified or approximated further if necessary.

Key Concepts

Radical Expressions

Radical expressions involve roots, like square roots, cube roots, etc. These expressions are pivotal in algebra when you deal with equations that include fractional powers.
They serve as a tool to express roots that may not have nice, round integer solutions.
In the equation given, the term \(x^{3/2}\) is a radical expression. The fractional exponent signifies a power and a root combined. Understanding radical expressions involves getting comfortable with:

• **Square Roots:** Notated as \(\sqrt{x}\) where \(x^{1/2}\) represents a square root of \(x\).
• **Cube Roots:** Written as \(\sqrt[3]{x}\); an expression such as \(x^{1/3}\) indicates the cube root.

To isolate radical expressions, the goal is to simplify the equation to get the radical on one side.
This often involves basic operations like addition, subtraction, and division as was done when isolating \(4x^{3/2}\) to \(x^{3/2} = 4\). Once isolated, the expression can be dealt with more straightforwardly.

Fractional Exponents

Fractional exponents, also known as rational exponents, are an innovative way to represent roots in algebra.
Instead of using radical signs, fractional exponents reveal both the power and the root through a single notation.
For example, in the expression \(x^{3/2}\), the fraction \(\frac{3}{2}\) indicates both a power and a root. Here, 3 is the power, and 2 is the root, meaning the same as taking the square root of \(x^3\).
Let's recall:

• **The Denominator as a Root:** The denominator specifies the root. \(x^{1/3}\) means the cube root of \(x\).
• **The Numerator as a Power:** The numerator signifies the power. \(x^{3/2}\) translates to the square root of \(x^3\).
• **Simplification Using Exponent Rules:** Apply the rule \((a^m)^n = a^{m \times n}\) to simplify equations.

In practice, converting radicals to fractional exponents can simplify many types of equations across algebra by working in the numeric form familiar to students.
Understanding these conversions and simplifications is crucial for solving equations involving nonlinear expressions.

Equation Solving

Solving equations is at the heart of algebra. This skill involves finding the values of variables that make an equation true.
It requires combining different mathematical operations to strategically simplify and solve for the unknown.
The primary steps to solve any equation typically include:

• **Isolating the Variable:** By moving terms around using basic operations (addition, subtraction, multiplication, and division) until the variable is by itself.
• **Applying Inverse Operations:** Use inverse operations to undo addition/subtraction and multiplication/division.
• **Simplifying Complex Equations:** Break down more complicated terms systematically, like converting fractional exponents to more manageable forms.

In our specific equation \(4x^{3/2} + 5 = 21\), solving involved isolating the fractional power term \(x^{3/2}\), then using exponent rules to further simplify.
Finally, calculating \(4^{2/3}\) achieved the exact solution for \(x\).
Each step leads incrementally toward making an equation simpler to find the variable's value.