What is the value of A such that `root (3)(500) xx root (3) (-3456)=40xx A` is true ?

Options

A. `-3`
B. `-12`
C. `-15`
D. `-2`

Answer

To find the value of A such that \( \sqrt[3]{500} \times \sqrt[3]{-3456} = 40 \times A \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sqrt[3]{500} \times \sqrt[3]{-3456} = 40 \times A \]
This can be rewritten as:
\[ \sqrt[3]{500 \times -3456} = 40 \times A \]

### Step 2: Calculate the product inside the cube root
Next, we need to calculate \( 500 \times -3456 \):
\[ 500 = 100 \times 5 = 10^2 \times 5 \]
\[ -3456 = -1 \times 3456 \]
Now, let's factor \( 3456 \):
\[ 3456 = 2^7 \times 3^3 \]
Thus, we have:
\[ 500 \times -3456 = (100 \times 5) \times (-1 \times 3456) = -500 \times 3456 \]

### Step 3: Simplify the cube root
Now we can compute:
\[ \sqrt[3]{-500 \times 3456} = \sqrt[3]{-1 \times 500 \times 3456} \]
Calculating \( 500 \times 3456 \):
\[ 500 \times 3456 = 100 \times 5 \times 3456 = 100 \times (5 \times 3456) \]
We can calculate \( 5 \times 3456 \):
\[ 5 \times 3456 = 17280 \]
Thus:
\[ 500 \times -3456 = -172800 \]

### Step 4: Calculate the cube root
Now we find:
\[ \sqrt[3]{-172800} \]
We can factor \( 172800 \):
\[ 172800 = 1728 \times 100 = 12^3 \times 10^2 \]
So:
\[ \sqrt[3]{-172800} = -\sqrt[3]{172800} = -12 \times 10^{2/3} \]

### Step 5: Set the equation equal to \( 40 \times A \)
Now we have:
\[ -12 \times 10^{2/3} = 40 \times A \]
To find A:
\[ A = \frac{-12 \times 10^{2/3}}{40} \]
Simplifying:
\[ A = \frac{-3 \times 10^{2/3}}{10} = -\frac{3}{10} \times 10^{2/3} \]

### Step 6: Final value of A
Thus, the value of A is:
\[ A = -\frac{3}{10} \times 10^{2/3} \]