Understanding the Algebraic Problem

The question asks us to find the cube of a specific number. We are given information about this number in the form of a word problem, which we can translate into an algebraic equation. The problem states that when 16 is subtracted from 3 times a number, the result is 8.

Let's represent the unknown number with a variable, say 'x'.

Setting Up the Equation

Based on the problem description, we can set up the equation as follows:

• "3 times a number": This can be written as \(3 \times x\) or simply \(3x\).
• "16 is subtracted from 3 times a number": This means we take \(3x\) and subtract 16 from it. So, it becomes \(3x - 16\).
• "The result is 8": This tells us that the expression \(3x - 16\) is equal to 8.

Putting it all together, the equation is:

\(3x - 16 = 8\)

Solving the Equation for the Number

Now, we need to solve this linear equation for the variable 'x' to find the original number. We want to isolate 'x' on one side of the equation.

1. Add 16 to both sides of the equation to get rid of the -16 on the left side:

   \(3x - 16 + 16 = 8 + 16\)

   \(3x = 24\)

2. Divide both sides of the equation by 3 to solve for 'x':

   \(\frac{3x}{3} = \frac{24}{3}\)

   \(x = 8\)

So, the original number is 8.

Calculating the Cube of the Original Number

The question asks for the cube of the original number. We found the original number to be 8. The cube of a number is that number multiplied by itself three times.

Cube of x is \(x^3\).

In our case, the number is 8, so we need to calculate \(8^3\).

\(8^3 = 8 \times 8 \times 8\)

First, calculate \(8 \times 8\):

\(8 \times 8 = 64\)

Now, multiply the result by 8 again:

\(64 \times 8\)

To calculate \(64 \times 8\):

• \(8 \times 4 = 32\). Write down 2, carry over 3.
• \(8 \times 6 = 48\). Add the carried over 3: \(48 + 3 = 51\).
• The result is 512.

So, \(8^3 = 512\).

Final Result

The original number is 8, and its cube is 512.

Revision Table: Solving Word Problems

Additional Information: Algebraic Equations and Exponents

This problem involves two key mathematical concepts: setting up and solving a linear algebraic equation, and calculating exponents (specifically, the cube of a number).

What is an Algebraic Equation?

An algebraic equation is a mathematical statement that shows that two expressions are equal. It contains variables (like 'x'), numbers, and mathematical operations (+, -, *, /). Solving an equation means finding the value(s) of the variable(s) that make the statement true.

Solving Linear Equations

A linear equation is an equation where the highest power of the variable is 1 (like \(3x\)). To solve linear equations, we use inverse operations to isolate the variable. The goal is to perform the same operation on both sides of the equals sign to maintain balance.

• If a number is subtracted, add it to both sides.
• If a number is added, subtract it from both sides.
• If a number is multiplied, divide both sides by it.
• If a number is divided, multiply both sides by it.

Understanding Cubes (Exponents)

The cube of a number is the result of multiplying the number by itself three times. It is represented by raising the number to the power of 3 (e.g., \(x^3\)). \(x^3 = x \times x \times x\). Calculating cubes is an operation involving exponents.