Problem 71 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \(\sqrt[3]{x}=5\)

Step-by-Step Solution

Answer

The real solution is \( x = 125 \).

Step 1: Understand the Given Equation

The equation \( \sqrt[3]{x} = 5 \) involves a cube root on the left side. Our goal is to find the value of \( x \) that satisfies this equation.

Step 2: Rewrite the Equation as a Power

To eliminate the cube root, rewrite the equation as \( x^{1/3} = 5 \). This represents \( x \) raised to the power of \( 1/3 \).

Step 3: Raise Both Sides to the Third Power

To solve for \( x \), raise both sides of the equation to the third power to undo the cube root: \[ (x^{1/3})^3 = 5^3 \].

Step 4: Simplify the Equation

Simplifying the equation from Step 3 gives \( x = 5^3 \). This step removes the cube root by applying the cube power.

Step 5: Calculate the Cubed Value

Compute \( 5^3 \), which is \( 5 \times 5 \times 5 = 125 \). This calculation gives us the value of \( x \).

Key Concepts

Cube Root

The concept of the cube root is a fundamental idea in mathematics. The cube root of a number is a value that, when multiplied by itself three times, results in the original number. For example, if you have a number \( x \), written as \( \sqrt[3]{x} \), it means finding the number that, when used in the multiplication \( n \times n \times n \), returns \( x \). In the given equation \( \sqrt[3]{x} = 5 \), we are looking for a value of \( x \) that equals its cube root, which provides simplicity in many situations where cube roots appear.Cube roots are closely related to exponents because taking the cube root is the same as raising a number to the power of \( 1/3 \). Understanding this connection helps when solving equations like \( \sqrt[3]{x} = 5 \) by turning the cube root into an exponent, making calculations and manipulations more intuitive.

Exponents

Exponents play a vital role in simplifying and solving equations. They represent repeated multiplication of a number by itself. In our equation, transitioning from a cube root to an exponent makes solving for \( x \) more straightforward. By rewriting \( \sqrt[3]{x} \) as \( x^{1/3} \), we convert the root into an easily manageable form using exponents.Generally, when handling roots and exponents, the following key rules are useful:

• \( a^{m/n} = \sqrt[n]{a^m} \)
• \( (a^m)^n = a^{m \times n} \)

In our example, raising both sides to the power of 3 eliminates the cube root, simplifying to \( x = 5^3 \). This simplification helps to calculate the value directly.Exponents make it easy to handle large numbers or break down roots into simpler, solvable numbers, making them essential in algebraic problem-solving.

Real Solutions

When solving an equation, the goal is often to find all of its real solutions. Real solutions refer to any solution that is a real number, which can be positive, negative, or zero—essentially, any number that exists on the infinite number line.In the example equation \( \sqrt[3]{x} = 5 \), finding the real solution involves determining \( x \) when the equation holds true. After rewriting the equation using exponents and solving, we find that \( x = 125 \) is the real solution.Real solutions are crucial in understanding the applicability of equations in real-world scenarios. When conducting experiments or making predictions, knowing that a real solution exists allows for concrete results and physical interpretations. Thus, identifying real solutions serves as a verification step, ensuring the solution is applicable beyond theoretical calculations, adding significant value to practical applications.