Solving for the Unknown: 3 × ? ÷ 9 = ∛729 × √121

This problem asks us to find the missing number represented by a question mark ('?') in a given mathematical equation. Let's break down the equation and solve it step-by-step.

Understanding the Equation

The equation is:

3 × ? ÷ 9 = ∛729 × √121

Our goal is to isolate the question mark ('?') and find its numerical value.

Step 1: Simplify the Right Side of the Equation

First, we need to calculate the values of the cube root and the square root:

• Cube Root of 729: We are looking for a number that, when multiplied by itself three times, equals 729. We can test numbers: $9 \times 9 \times 9 = 81 \times 9 = 729$. So, the cube root of 729 is 9. Mathematically, this is written as:

  \(\sqrt[3]{729} = 9\)

• Square Root of 121: We need to find a number that, when multiplied by itself, equals 121. We know that $11 \times 11 = 121$. So, the square root of 121 is 11. Mathematically, this is written as:

  \(\sqrt{121} = 11\)

Now, we multiply these two results:

Right Side = 9 × 11 = 99

Step 2: Simplify the Left Side of the Equation

Let's represent the unknown value (the question mark '?') with the variable 'x'. The left side of the equation becomes:

3 × x ÷ 9

This can be written as a fraction:

\(\frac{3 \times x}{9}\)

We can simplify this fraction by dividing both the numerator and the denominator by 3:

\(\frac{3x}{9} = \frac{x}{3}\)

Step 3: Equate Both Sides and Solve for x

Now we set the simplified left side equal to the simplified right side:

\(\frac{x}{3} = 99\)

To find the value of x, we multiply both sides of the equation by 3:

\(x = 99 \times 3\)

Calculating the final value:

\(x = 297\)

Conclusion

The number that should replace the question mark ('?') in the equation is 297.