Problem 11 - Intermediate Algebra

Question

Solve. \(\sqrt[3]{6 x}=-3\)

Step-by-Step Solution

Answer

\( x = -\frac{9}{2} \)

Step 1: Understanding the Equation

The equation we are working with is \( \sqrt[3]{6x} = -3 \). This indicates the cube root of \( 6x \) is equal to \( -3 \). Our goal is to solve for \( x \).

Step 2: Eliminate the Cube Root

To eliminate the cube root, we need to cube both sides of the equation. This means raising both sides to the power of 3: \((\sqrt[3]{6x})^3 = (-3)^3\).

Step 3: Simplify the Equation

Cubing \( \sqrt[3]{6x} \) gives \( 6x \), and cubing \( -3 \) gives \( -27 \). Thus, the equation simplifies to \( 6x = -27 \).

Step 4: Solve for x

To solve for \( x \), we divide both sides of the equation by 6, yielding \( x = \frac{-27}{6} \). Simplifying, \( x = -\frac{9}{2} \).

Key Concepts

Solving for x

When we solve for \( x \) in an equation, our goal is to find the value of \( x \) that makes the equation true. In the given exercise \( \sqrt[3]{6x} = -3 \), the cube root indicates that to isolate \( x \), we need to perform a series of operations. The first step is to eliminate the cube root by cubing both sides of the equation. This is because cubing a cube root cancels it out, leaving the expression inside the root by itself.

After cubing, the equation \( 6x = -27 \) results, making it much simpler. Now, we need to isolate \( x \). This is done by dividing both sides of the equation by 6, as \( x \) is currently being multiplied by 6.

• Divide left and right by 6: \( x = \frac{-27}{6} \)
• Simplifying the fraction gives: \( x = -\frac{9}{2} \)

By finding that \( x = -\frac{9}{2} \), we've successfully solved for \( x \), making sure our operations maintain equality and simplify the path to the solution.

Simplifying Equations

Simplifying equations involves making them easier to work with by reducing complexity. In the exercise, we start by removing the cube root from \( \sqrt[3]{6x} = -3 \) since it complicates direct solving for \( x \). We achieve this by cubing both sides, allowing us to transform the equation into a linear form \( 6x = -27 \).

This transformation is indispensable for simplification because it eliminates the root, and we now have straightforward arithmetic to deal with, rather than dealing with radicals. Simplification doesn’t end here. When solving for \( x \), dividing \( -27 \) by 6 gives a fraction \( x = \frac{-27}{6} \).

• Reduce fractions by finding the greatest common divisor (GCD) of the numerator and the denominator.
• For \( \frac{-27}{6} \), the GCD is 3. Thus, reduced: \( x = -\frac{9}{2} \)

Simplifying helps not only in finding exact values but also in understanding what's happening in each step of the solution.

Cube Root Properties

Cube root properties are useful in manipulating equations where cube roots are involved. The cube root of a number \( a \) is a value \( b \) such that \( b^3 = a \). It has unique characteristics that differ from square roots, one being that they can have negative results, as in our exercise \( \sqrt[3]{6x} = -3 \). Unlike square roots, cube roots of negative numbers are real numbers, and this is crucial in handling equations with them.

To eliminate a cube root, we apply the cubic operation to both sides. This step is reversible because of the innate property \( (\sqrt[3]{a})^3 = a \). Understanding cube root properties helps

• Recognize that cubing both sides is an inverse operation that will yield \( a = b \) when \( \sqrt[3]{a} = b \).
• Work with negative values effortlessly because cube roots can handle them unlike square roots, where negative values under the root create imaginary numbers.

Mastering these characteristics helps you solve equations involving cube roots confidently.