Problem 69 - College Algebra with Modeling and Visualization

Question

Solve the equation. Check your answers. $$ 4 x^{3 / 2}+5=21 $$

Step-by-Step Solution

Answer

The solution is \( x = 4^{2/3} \), which satisfies the equation.

Step 1: Isolate the term with the variable

To solve the equation, first subtract 5 from both sides to isolate the term with the variable. This gives us: \[ 4x^{3/2} = 16 \]

Step 2: Divide by the coefficient of the variable

Divide both sides of the equation by 4 to solve for the expression with the power: \[ x^{3/2} = 4 \]

Step 3: Eliminate the fractional exponent

Raise both sides to the power of \( \frac{2}{3} \) to get rid of the fractional exponent on \( x \). This gives us: \[ x = 4^{2/3} \]

Step 4: Simplify the expression

Calculate \( 4^{2/3} \) by finding the cube root of 4, then squaring the result. The cube root of 4 is \( \sqrt[3]{4} \), which can be simplified by leaving it as is for an exact form or calculating approximately using a calculator.

Step 5: Verify the solution

Substitute \( x = 4^{2/3} \) back into the original equation to check whether it satisfies the equation: \[ 4(4^{2/3})^{3/2} + 5 \] Simplify: \[ 4 \times 4 + 5 = 21 \] This confirms that \( x = 4^{2/3} \) satisfies the equation.

Key Concepts

Fractional Exponents

Understanding fractional exponents is crucial in algebra, as they allow us to express roots in a different form. Typically, an exponent is a way to represent repeated multiplication. However, when we have fractional exponents, it involves both a root and a power. Consider the expression \( x^{3/2} \):

• The number in the numerator (3) indicates the power to which the base \( x \) is raised.
• The denominator (2) indicates the root we need to take, which in this case, is the square root.

Thus, \( x^{3/2} \) can be read as "the square root of \( x^3 \)," or conversely, "\((x^{1/2})^3\)." Learning to switch from radical notation to fractional exponents and vice versa increases your flexibility when manipulating equations.
In our original exercise, solving for \( x \) required us to eliminate the fractional exponent by taking another power, specifically \( \frac{2}{3} \). This step effectively cancels the \( \frac{3}{2} \) exponent on \( x \), leaving us with the base alone.

Fractional exponents simplify many algebraic operations, making it easier to manage complex expressions.

Isolation of Variables

When solving equations, one of the first steps is usually isolating the variable of interest. This means rearranging the equation so that the variable stands alone on one side. Isolating the variable simplifies the problem, allowing us to see the path to solution.
In the exercise, we start by isolating the term with the variable \( x \) by eliminating constant terms and coefficients from the equation. Here’s how we did it in our example:

• Subtract 5 from both sides: \(4x^{3/2} + 5 = 21\) becomes \(4x^{3/2} = 16\).
• Next, divide both sides by 4 to remove the coefficient in front of \( x \): \(x^{3/2} = 4\).

By isolating the variable, we simplify the equation from a complex one to something manageable with exposure to fractional exponents, bringing us closer to the solution.

Checking Solutions

After finding a solution to an algebraic equation, it is always good practice to verify that your solution is correct. This supports the integrity of your result and ensures you've followed the right steps without error.
To check your solution, substitute the value you found back into the original equation and see if both sides balance. For example, with our equation:

• We found that \(x = 4^{2/3}\).
• Substitute \(x\) back: \(4(4^{2/3})^{3/2} + 5\).
• Simplify the expression to confirm that it equals 21, as in the original equation.

By checking the solution, you confirm the accuracy of not only that solution, but also your overall problem-solving process. This step is vital, especially in more complex exercises where errors can easily occur.