Problem 57 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{3}=27 $$

Step-by-Step Solution

Answer

The real solution to the equation \(x^3 = 27\) is \(x = 3\).

Step 1: Identify the Equation Type

The given equation is \( x^3 = 27 \), which is a cubic equation. Our task is to find the real solutions of this equation. Since it involves a single variable raised to the power of 3, it is essentially finding the cube root of a number.

Step 2: Isolate the Variable

To solve for \( x \), consider the equation \( x^3 = 27 \). We need to isolate \( x \), which can be done by taking the cube root of both sides of the equation. Therefore, we compute \( x = \sqrt[3]{27} \).

Step 3: Calculate the Cube Root

Calculate the cube root of 27. Since 27 is a perfect cube, \( 27 = 3^3 \), the cube root is straightforward: \( \sqrt[3]{27} = 3 \).

Step 4: Verify the Solution

Verify that \( x = 3 \) is indeed a solution by substituting back into the original equation. Calculate \( 3^3 \), which gives \( 27 \), confirming that our solution is correct because it satisfies the original equation.

Key Concepts

Real Solutions

In mathematics, real solutions are those values for the variable that satisfy the equation and fall within the set of real numbers. Real numbers include both rational and irrational numbers, covering all values that you can find on the number line that aren't imaginary. In the context of solving equations, finding real solutions means isolating the variable and determining which values make the equation true.For a cubic equation like \( x^3 = 27 \), the goal is to find a real number that, when cubed, equals 27. Since the result is a positive number, the cube root will also be positive in this particular example. Furthermore, cubic equations can have up to three real solutions, but in this simple equation of \( x^3 = 27 \), there is only one real solution. This simplifies our task to identifying the cube root of 27 to find our answer.

Cube Root Calculation

Calculating the cube root of a number is a key step in solving cubic equations. In essence, you are finding the number that, when multiplied by itself twice more, gives you the original number. For the equation \( x^3 = 27 \), we need to find the cube root of 27 to solve for \( x \).Since 27 is a perfect cube (as it can be expressed as \( 3^3 \)), finding the cube root is straightforward. The cube root of 27 is 3, which means \( \sqrt[3]{27} = 3 \). This is because \( 3 \times 3 \times 3 = 27 \). Understanding how to identify and calculate cube roots is helpful for swiftly solving such equations.

Equation Solving Steps

Solving equations effectively involves a systematic approach. Let's look at how to solve the cubic equation \( x^3 = 27 \) using defined steps.

• First, identify that the equation you are dealing with is a cubic equation. This clues you into the fact you're seeking the cube root of a certain number.
• Next, isolate the variable. For our equation, \( x^3 = 27 \), this means resolving to the form \( x = \sqrt[3]{27} \). This isolates \( x \) on one side of the equation.
• Calculate the cube root. From the previous step, determine that \( \sqrt[3]{27} = 3 \), which gives us \( x = 3 \).
• Finally, verify your solution. Substitute \( x = 3 \) back into the original equation to check your work: \( 3^3 = 27 \). Since both sides equal each other, \( x = 3 \) is confirmed as the correct real solution.

Following a structured procedure as outlined ensures solving cubic equations remains straightforward while reducing potential errors.