Problem 108 - College Algebra

Question

Solve for \(x: \sqrt[3]{x \sqrt{x}}=9\)

Step-by-Step Solution

Answer

After calculating the cubic root of 531441, the solution for \( x \) is 81.

Step 1: Eliminate the cubic root

To remove the cubic root we cube both sides of the equation. So, we get \( (\sqrt[3]{x \sqrt{x}})^3 = 9^3 \), which simplifies to \( x\sqrt{x} = 729 \)

Step 2: Squaring both sides

To get rid of the square root, we square both sides of the equation. Then the equation becomes \( (x\sqrt{x})^2 = 729^2 \), which simplifies to \( x^3 = 531441 \)

Step 3: Solve the cubic equation

To find \( x \), we take the cubic root of both sides. Thus, \( x = \sqrt[3]{531441} \).

Key Concepts

Understanding Cubic Roots

Cubic roots are the inverse operation of cubing a number. When you encounter a cubic root in an equation, it means you're finding a number that, when multiplied by itself twice, will result in the given number.
For example, if you have a number, say 8, and you want to find its cubic root, you're looking for a number that satisfies this equation: \( n^3 = 8 \). The answer would be 2 because \( 2 \times 2 \times 2 = 8 \).
In the context of equations, cubic roots help us solve problems where the variable is raised to the power of three. To isolate a variable under a cubic root, we often need to cube both sides of an equation, which effectively "undoes" the cubic root, simplifying the problem.

Exploring Exponential Equations

Exponential equations involve variables that appear as exponents. These equations can be challenging because the variable isn't just multiplied or added - it's part of a power.
An example of an exponential equation is \( 2^x = 64 \). To solve such equations, you often want to express each side of the equation as powers of the same base. In this case, both sides can be rewritten as powers of 2: \( 2^x = 2^6 \).
Once that's done, because the base on both sides is the same, you can set the exponents equal to each other, so \( x = 6 \). This method makes solving exponential equations manageable even when direct manipulation seems complex.

Techniques for Solving Equations

Solving equations is about finding the value of the variable that makes the equation true. Different types of equations require different techniques and strategies.

• Isolating the Variable: Start by moving all terms with the variable to one side of the equation, and all constant terms to the other. This might involve addition, subtraction, multiplication, or division.
• Using Roots and Powers: If the equation involves quadratic, cubic, or higher powers, you might use roots (like cubic roots) or powers to simplify the expression.
• Factoring: For polynomial equations, factoring can be useful. Identify common factors or use special formulas to simplify the equation into solvable parts.

These strategies provide a toolkit for approaching different equations logically and systematically. Making sure to perform the same operations on both sides maintains the equation's balance and makes finding the solution clearer.