Problem 52 - College Algebra

Question

(Hint: To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \((\sqrt[n]{y})^{n}=y .\) ) $$ f(x)=\sqrt[3]{x-1} $$

Step-by-Step Solution

Answer

The solution for x in the given equation is \(x=y^3+1\)

Step 1: Rewrite Equation

The equation to solve is \(f(x)=\sqrt[3]{x-1}\). Assuming \(f(x) = y\) we have: \(y=\sqrt[3]{x-1}\).

Step 2: Remove Cube Root

To remove the cube root from the equation, raise both sides of the equation to the power of 3. This is due to the property that \((\sqrt[n]{x})^{n}=x\). Following this rule we can write: \(y^3=(\sqrt[3]{x-1})^3\).

Step 3: Solve for x

After raising both sides of the equation to the power of 3, we get \(y^3=x-1\). From here, we can add 1 to both sides of the equation to isolate x. Final expression will be \(x=y^3+1\).

Key Concepts

Cube Root Properties

Understanding the properties of cube roots is essential for solving equations that include them. The cube root of a number, \textbf{written as} \( \sqrt[3]{x} \), is a value that, when raised to the third power, gives the original number. In other words, if \( a = \sqrt[3]{b} \), then it is true that \( a^3 = b \
\
\). This is the foundational property used to eliminate cube roots when solving equations.

Another important property to recall is that the cube root of a negative number is also negative, as opposed to square roots which are not defined for negative numbers in the real number system. This means that you can solve cube root equations over all real numbers.

When you encounter a cube root in an equation, the goal is often to isolate this root and then raise both sides of the equation to the third power to remove the root, simplifying the equation for easier solution.

Raising to a Power

When you raise a number to a power, you're multiplying that number by itself a certain number of times. For instance, \( x^3 \) means \( x \times x \
\
\times x \). When dealing with variables under a radical, like a cube root, this concept becomes particularly useful. To eliminate a cube root, you can raise both sides of an equation to the third power. This is based on the principle that the operations are inverses of each other—\( (\sqrt[3]{x})^3 = x \).

However, it's important to note that precision is key when raising exponents to a power. Make sure that the entire cube root expression is raised to the third power, not just the variable inside. For example, to correctly remove the cube root in \( \sqrt[3]{x-1} \), you would raise the entire expression to the third power, resulting in \( (\sqrt[3]{x-1})^3 = x-1 \
\
\). Paying close attention to these details will prevent common mistakes in solving equations that involve powers.

Isolating Variables

Isolating the variable in an equation means rearranging the equation so that the variable is on one side by itself. It's a fundamental technique in algebra to solve for unknowns. To achieve isolation, you use inverse operations that counteract whatever operation is being performed on the variable.

For instance, if a variable is being added to a number, subtract that number from both sides. If a variable is under a cube root, you eliminate the root by raising both sides to the third power, as in the previous examples. Once the variable is free from other numbers or operations, you can find its value by performing the same operations on both sides of the equation.

With the earlier example \( y = \sqrt[3]{x-1} \), by raising both sides to the third power, you isolate \( x-1 \) on one side, giving \( y^3 = x-1 \). You can then add 1 to both sides to complete the isolation: \( x = y^3 + 1 \
\
\). This technique is vital for making complex equations simple and solvable.