Question

\\j^3 = 1,728\\
\\w^3 = 27,000\\

Explanation:

Identify the equations

We need to solve two separate cubic equations for the variables \(j\) and \(w\):

1. \(j^3 = 1,728\)
2. \(w^3 = 27,000\)

Solve the first equation

To find \(j\), we take the cube root of both sides:

$$j = \sqrt[3]{1,728}$$

We look for an integer that, when multiplied by itself three times, equals \(1,728\).
Since \(10^3 = 1,000\) and \(20^3 = 8,000\), the number is between \(10\) and \(20\).
Since the last digit is \(8\), and \(2^3 = 8\), we test \(12\):

$$12 \times 12 \times 12 = 144 \times 12 = 1,728$$

Thus, \(j = 12\).

Solve the second equation

To find \(w\), we take the cube root of both sides:

$$w = \sqrt[3]{27,000}$$

We can simplify this by factoring:

$$27,000 = 27 \times 1,000$$

Taking the cube root of each factor:

$$\sqrt[3]{27} = 3$$

$$\sqrt[3]{1,000} = 10$$

Multiplying these results together:

$$w = 3 \times 10 = 30$$

Answer:

Question 1

\(j = 12\)

Question 2

\(w = 30\)