Question

Which product yields an irrational number? A. sqrt(9) * 3 B. 7/8 * sqrt(25) C. 2sqrt(64) * 7sqrt(81) D. 4sqrt(5) * 2sqrt(49)

Answer 1

Let's analyze each option to determine if it yields an irrational number:

**A.** \( \sqrt{9} \cdot 3 \)

Calculating \( \sqrt{9} \):
\[
\sqrt{9} = 3
\]
Thus,
\[
\sqrt{9} \cdot 3 = 3 \cdot 3 = 9
\]
This is a rational number.

**B.** \( \frac{7}{8} \cdot \sqrt{25} \)

Calculating \( \sqrt{25} \):
\[
\sqrt{25} = 5
\]
Thus,
\[
\frac{7}{8} \cdot 5 = \frac{35}{8}
\]
This is also a rational number.

**C.** \( 2\sqrt{64} \cdot 7\sqrt{81} \)

Calculating \( \sqrt{64} \) and \( \sqrt{81} \):
\[
\sqrt{64} = 8 \quad \text{and} \quad \sqrt{81} = 9
\]
Thus,
\[
2\sqrt{64} = 2 \cdot 8 = 16
\]
\[
7\sqrt{81} = 7 \cdot 9 = 63
\]
Now calculating \( 16 \cdot 63 \):
\[
16 \cdot 63 = 1008
\]
This is a rational number.

**D.** \( 4\sqrt{5} \cdot 2\sqrt{49} \)

Calculating \( \sqrt{49} \):
\[
\sqrt{49} = 7
\]
Thus,
\[
2\sqrt{49} = 2 \cdot 7 = 14
\]
Now calculating \( 4\sqrt{5} \cdot 14 \):
\[
4\sqrt{5} \cdot 14 = 56\sqrt{5}
\]
Since \( \sqrt{5} \) is an irrational number, \( 56\sqrt{5} \) is also irrational.

After analyzing all options, the only choice that yields an irrational number is:
**D. \( 4\sqrt{5} \cdot 2\sqrt{49} \)**.