Problem 1 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ x^{3}=50 $$

Step-by-Step Solution

Answer

Answer: The value of x in the equation \(x^3 = 50\) is approximately 3.684.

Step 1: Understand the equation

The given equation is \(x^3 = 50\). We need to solve for x by finding the value that, when cubed, results in 50.

Step 2: Find the cube root

To solve for x, we need to find the cube root of 50. The cube root of a number is the value that, when multiplied by itself three times, gives that number. We can represent the cube root using the "∛" symbol or as a fractional exponent. Thus, we have two equivalent expressions for calculating the cube root of 50: $$ x = \sqrt[3]{50} $$ or $$ x = 50^{\frac{1}{3}} $$

Step 3: Calculate the cube root

Use a calculator to find the cube root of 50: $$ x = \sqrt[3]{50} \approx 3.684 $$ or $$ x = 50^{\frac{1}{3}} \approx 3.684 $$

Step 4: Write the final solution

Since the cube root of 50 cannot be simplified further, the final solution is: $$ x \approx 3.684 $$

Key Concepts

Cube Root

The cube root of a number is essential when you want to undo a cubed number. Think of it like finding the opposite of cubing a number. If you take any number and multiply it by itself twice (meaning three times in total), you will get a bigger number. However, if you start with the bigger number and want to get back to your original number, you would take the cube root.
For instance:

• The cube root of 27 is 3 because 3 × 3 × 3 = 27.
• You can express the cube root using the symbol ∛. For our example, it's written as ∛27 = 3.

But what if your number isn't a nice, neat cube like 27? That's where we use calculators. As in our exercise, the number 50 isn’t a neat cube like 27, so we use the expression \(x = 50^{\frac{1}{3}}\) to determine the cube root which is approximately 3.684.

Exponents

Exponents are used to denote the number of times a number is multiplied by itself. In the context of our equation, we start with the expression \(x^3 = 50\). The "3" here is the exponent, indicating that "x" is being multiplied by itself twice more (a total of three times).
Exponents are powerful tools:

• They simplify expressions that would otherwise require lengthy multiplication.
• You can use fractional exponents to express roots; for instance, \(x^{\frac{1}{3}}\) represents the cube root of x.

In our exercise, moving from \(x^3 = 50\) to finding \(x = 50^{\frac{1}{3}}\) showcases changing from a whole number exponent to a fractional one to help us solve the equation.

Algebraic Solutions

Algebraic solutions involve manipulating an equation to find the value of a variable. Solving equations allows us to find unknowns using logical operations and principles. In the case of \(x^3 = 50\), we aim to isolate "x" and find its value. Here's how the process unfolds:

• The task: Find "x" such that when cubed, it equals 50.
• Step: Recognize that taking the cube root will isolate x. Thus, x is expressed as \(\sqrt[3]{50}\) or \(50^{\frac{1}{3}}\).
• Outcome: Use a calculator for precision, resulting in \(x \approx 3.684\).

Through this process, we logically work through the equation to arrive at a solution that makes sense numerically and algebraically. Always remember, understanding the operations and manipulations is crucial to solving algebraic equations effectively.