Question

drag each expression into the correct category.
result is a rational number result is an irrational number
$7 + \sqrt{5}$ $7\sqrt{5}$ $7 + 5$ $7(5)$ $\sqrt{7} + \sqrt{5}$ $\sqrt{7} \cdot \sqrt{5}$ $\sqrt{7} + \sqrt{7}$ $\sqrt{7} \cdot \sqrt{7}$

Explanation:

Step1: Identify rational/irrational for $7+\sqrt{5}$

$\sqrt{5}$ is irrational; rational + irrational = irrational.

Step2: Identify rational/irrational for $7\sqrt{5}$

$\sqrt{5}$ is irrational; rational $\times$ irrational = irrational.

Step3: Identify rational/irrational for $7+5$

$7+5=12$, integer (rational number).

Step4: Identify rational/irrational for $7(5)$

$7(5)=35$, integer (rational number).

Step5: Identify rational/irrational for $\sqrt{7}+\sqrt{5}$

Sum of two distinct irrationals = irrational.

Step6: Identify rational/irrational for $\sqrt{7} \cdot \sqrt{5}$

$\sqrt{7 \times 5}=\sqrt{35}$, irrational number.

Step7: Identify rational/irrational for $\sqrt{7}+\sqrt{7}$

$\sqrt{7}+\sqrt{7}=2\sqrt{7}$, rational $\times$ irrational = irrational.

Step8: Identify rational/irrational for $\sqrt{7} \cdot \sqrt{7}$

$\sqrt{7} \cdot \sqrt{7}=7$, integer (rational number).

Answer:

Result is a Rational Number:

$7+5$, $7(5)$, $\sqrt{7} \cdot \sqrt{7}$

Result is an Irrational Number:

$7+\sqrt{5}$, $7\sqrt{5}$, $\sqrt{7}+\sqrt{5}$, $\sqrt{7} \cdot \sqrt{5}$, $\sqrt{7}+\sqrt{7}$