To find the product of \( 2\sqrt{4} \) and \( 5\sqrt{13} \), we first simplify each term.
-
Simplify \( 2\sqrt{4} \): \[ \sqrt{4} = 2 \] Thus, \[ 2\sqrt{4} = 2 \cdot 2 = 4 \]
-
Now, we calculate \( 5\sqrt{13} \): \[ 5\sqrt{13} \text{ is already in its simplest form.} \]
Next, we find the product of the two simplified expressions: \[ 4 \cdot (5\sqrt{13}) = 4 \cdot 5\sqrt{13} = 20\sqrt{13} \]
Next, we need to determine if \( 20\sqrt{13} \) is rational or irrational.
- The number \( 20 \) is a rational number.
- The square root \( \sqrt{13} \) is known to be irrational, as \( 13 \) is not a perfect square.
The product of a non-zero rational number and an irrational number is always irrational. Therefore, since \( \sqrt{13} \) is irrational, \( 20\sqrt{13} \) must also be irrational.
Summary
The product of \( 2\sqrt{4} \) and \( 5\sqrt{13} \) is: \[ 20\sqrt{13} \] This result is irrational because it is the product of a rational number (20) and an irrational number (\(\sqrt{13}\)).