Problem 24 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The solution to the equation is approximately \( x \approx 2.52 \).

Step 1: Isolate the Term with the Variable

Start by subtracting 5 from both sides of the equation to isolate the term containing the variable \( x \). This gives us:\[ 4x^{3/2} = 16 \]

Step 2: Divide Both Sides by 4

Divide both sides of the equation by 4 to further simplify:\[ x^{3/2} = 4 \]

Step 3: Remove the Fractional Exponent

To solve for \( x \), eliminate the exponent by raising both sides to the power of \( \frac{2}{3} \). This will leave us with:\[ x = 4^{2/3} \]

Step 4: Simplify \(4^{2/3}\)

Evaluate \( 4^{2/3} \). Start by writing \( 4 \) as \( 2^2 \), and then apply the exponent:\[ (2^2)^{2/3} = 2^{4/3} = (2^{4/3}) \]This results in: \[ x = 2^{4/3} \approx 2.52 \] Thus, for simplicity, we find two parts first:\( 2^2 = 4 \) and\( 4^{1/3} \approx 1.587 \) by taking cube root of 4.Finally,\( x \approx 2.52 \).

Key Concepts

Fractional Exponents

Fractional exponents are a way to express numbers that involve both roots and powers. Instead of writing a square root or cubic root, we can use exponents in fraction form. For instance, the notation \( x^{3/2} \) can be understood in two steps:

• The number is first squared (since 3/2 = 1.5, which can be seen as 3 divided by 2).
• Then, the square root is applied (since it involves the division by 2 part).

Here's a simple breakdown to remember:

• If you see \( x^{n/m} \), it means "take the m-th root of \( x \) and then raise it to the n-th power."
• If you flip it to \( (x^n)^{1/m} \), it's the same. First raise \( x \) to the n-th power, then take the m-th root.

Fractional exponents make calculations more systematic and can simplify solving equations that involve complex roots.

Equation Isolation

Equation isolation is about simplifying an equation to focus on the part containing the variable. By isolating the variable, we change a complex equation into a simpler one that is easier to solve. Let's understand how this works:

• Normally, in an equation, multiple elements can be tied to our variable term (like numbers added to or multiplied with it).
• The goal is to "isolate" the variable term on one side of the equation.
• This often involves performing operations like addition, subtraction, multiplication, or division on both sides to keep the equation balanced.

In our exercise, we isolated \( 4x^{3/2} \) by first subtracting 5 from both sides and then dividing by 4. This step simplifies the problem significantly by focusing on the power expression \( x^{3/2} = 4 \), readying it for further manipulation.

Simplification Steps

Simplification steps help in reducing complex expressions into more manageable forms. They involve rewriting expressions to make solving equations easier. Here’s how to apply them effectively:

• Once you have an isolated variable term with a fractional exponent, like \( x^{3/2} = 4 \), removing the exponent is crucial. Here, exponentiation can be reversed by applying the reciprocal power.
• In our case, raise both sides to the power of \( \frac{2}{3} \) to cancel out the fractional exponent on \( x \).
• The next part involves simplifying any remaining terms. For instance, \( 4^{2/3} \) translates to taking the cube root first (since it is to the power of \( \frac{1}{3} \)), then squaring the result.

The true goal of simplification is accuracy and clarity, leading us closer to the answer in a swift, understandable series of steps. Each simplification brings us nearer to the final result, making the complex appear deceptively simple.