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Chapter 6: Problem 52

Solve. $$ 375=24 x^{3} $$

Short Answer

Expert verified
x = 2.5

Step by step solution

01

Isolate the Variable Term

First, isolate the term that includes the variable. The equation is given as 375 = 24x^3. Divide both sides of the equation by 24 to isolate the term involving x.
02

Simplify

After dividing both sides by 24, you get \[x^3 = \frac{375}{24}\]. Simplify the fraction. \[x^3 = 15.625\].
03

Take the Cube Root

To solve for x, take the cube root of both sides of the equation. \[x = \sqrt[3]{15.625}\].
04

Calculate the Cube Root

\[x = 2.5\] since \(2.5^3 = 15.625\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variable Terms
Isolating variable terms is a key step in solving equations. It means getting the variable or variable term by itself on one side of the equation. To do this, you often need to 'undo' other operations. In our exercise, we begin with the equation: \[375 = 24x^3\].
To isolate the variable term, we divide both sides by 24: \[ \frac{375}{24} = x^3\].
Dividing both sides helps to isolate the term involving x. This step is crucial because once we isolate the variable term, we can then focus on solving for the variable through other methods.
Simplifying Fractions
Simplifying fractions is an important skill in algebra. When you simplify fractions, you reduce them to their simplest form, making them easier to work with. In our equation, after isolating the variable term, we have: \[x^3 = \frac{375}{24}\].
Let's simplify this fraction. Break down both the numerator (375) and the denominator (24) into their prime factors:
  • 375 = 3 脳 5 脳 5 脳 5 = 3 脳 5^3
  • 24 = 2 脳 2 脳 2 脳 3 = 2^3 脳 3

Notice that both the numerator and the denominator have a common factor of 3. So, we divide both by 3: \[ \frac{375}{24} = \frac{3 脳 5^3}{2^3 脳 3} = \frac{5^3}{2^3} = \frac{125}{8}\].
Now we have the simplified form of the fraction: \[x^3 = 15.625\].
Cube Roots
Taking the cube root is how we find the value of the variable when it is raised to the power of 3. In our example, our simplified equation is: \[x^3 = 15.625\].
To solve for x, we take the cube root of both sides of the equation: \[x = \sqrt[3]{15.625}\]. The cube root of 15.625 is 2.5 because: \[2.5 \times 2.5 \times 2.5 = 15.625\].
Therefore, the solution to the original equation 375 = 24x^3 is: \[x = 2.5\].
Understanding cube roots is important as it helps to unravel the variable from an exponent of 3, providing a straightforward solution.

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