If `z_(1)=2 + 7i and z_(2)=1- 5i`, then verify that
`|z_(1)z_(2)|= |z_(1)||z_(2)|`

Answer

To verify that \( |z_1 z_2| = |z_1| |z_2| \) for the given complex numbers \( z_1 = 2 + 7i \) and \( z_2 = 1 - 5i \), we will follow these steps:

### Step 1: Calculate \( z_1 z_2 \)

We start by multiplying the two complex numbers:

\[ z_1 z_2 = (2 + 7i)(1 - 5i) \]

Using the distributive property (FOIL method):

\[ = 2 \cdot 1 + 2 \cdot (-5i) + 7i \cdot 1 + 7i \cdot (-5i) \]
\[ = 2 - 10i + 7i - 35i^2 \]

Since \( i^2 = -1 \), we have:

\[ -35i^2 = 35 \]

Now, combining the real and imaginary parts:

\[ = 2 + 35 - 10i + 7i \]
\[ = 37 - 3i \]

### Step 2: Calculate \( |z_1 z_2| \)

Now, we find the modulus of \( z_1 z_2 \):

\[ |z_1 z_2| = |37 - 3i| = \sqrt{37^2 + (-3)^2} \]
\[ = \sqrt{1369 + 9} = \sqrt{1378} \]

### Step 3: Calculate \( |z_1| \)

Next, we calculate the modulus of \( z_1 \):

\[ |z_1| = |2 + 7i| = \sqrt{2^2 + 7^2} \]
\[ = \sqrt{4 + 49} = \sqrt{53} \]

### Step 4: Calculate \( |z_2| \)

Now, we calculate the modulus of \( z_2 \):

\[ |z_2| = |1 - 5i| = \sqrt{1^2 + (-5)^2} \]
\[ = \sqrt{1 + 25} = \sqrt{26} \]

### Step 5: Calculate \( |z_1| |z_2| \)

Now we find the product of the moduli:

\[ |z_1| |z_2| = \sqrt{53} \cdot \sqrt{26} = \sqrt{53 \cdot 26} \]

Calculating \( 53 \cdot 26 \):

\[ 53 \cdot 26 = 1378 \]

Thus,

\[ |z_1| |z_2| = \sqrt{1378} \]

### Step 6: Verify the Equality

Now we can compare both sides:

\[ |z_1 z_2| = \sqrt{1378} \quad \text{and} \quad |z_1| |z_2| = \sqrt{1378} \]

Since both sides are equal, we have verified that:

\[ |z_1 z_2| = |z_1| |z_2| \]

### Conclusion

Thus, we have successfully verified that \( |z_1 z_2| = |z_1| |z_2| \).

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