Question

for each sum or product, determine whether the result is a rational number or an irrational number. then choose the appropriate reason for each.

	result is rational	result is irrational	reason
(a) $-\sqrt{19} + 5$	$\circ$	$\circ$	select
(b) $14 + 2.86$	$\circ$	$\circ$	select
(c) $10 \times \sqrt{15}$	$\circ$	$\circ$	select
(d) $-\sqrt{5} \times \sqrt{22}$	$\circ$	$\circ$	select

Explanation:

Part (a): \(-\sqrt{19} + 5\)

Step 1: Identify Number Types

\(\sqrt{19}\) is irrational (since 19 is not a perfect square). A rational number (5) plus an irrational number is irrational.

Step 2: Choose Result and Reason

Result is Irrational. Reason: Sum of a rational and an irrational number is irrational.

Part (b): \(14 + 2.86\)

Step 1: Identify Number Types

14 is an integer (rational), 2.86 is a terminating decimal (rational). Sum of two rational numbers is rational.

Step 2: Choose Result and Reason

Result is Rational. Reason: Sum of two rational numbers is rational.

Part (c): \(10 \times \sqrt{15}\)

Step 1: Identify Number Types

10 is rational, \(\sqrt{15}\) is irrational (15 is not a perfect square). Product of a non - zero rational and an irrational number is irrational.

Step 2: Choose Result and Reason

Result is Irrational. Reason: Product of a non - zero rational and an irrational number is irrational.

Part (d): \(-\sqrt{5} \times \sqrt{22}\)

Step 1: Simplify the Product

Using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), we get \(-\sqrt{5\times22}=-\sqrt{110}\). Since 110 is not a perfect square, \(\sqrt{110}\) is irrational, so \(-\sqrt{110}\) is irrational. Also, product of two irrational numbers ( \(\sqrt{5}\) and \(\sqrt{22}\)) is irrational (when the product is not a perfect square).

Step 2: Choose Result and Reason

Result is Irrational. Reason: Product of two irrational numbers (or simplifying gives an irrational number) is irrational.

Answer:

(a) Result is Irrational; Reason: Sum of a rational and an irrational number is irrational.
(b) Result is Rational; Reason: Sum of two rational numbers is rational.
(c) Result is Irrational; Reason: Product of a non - zero rational and an irrational number is irrational.
(d) Result is Irrational; Reason: Product of two irrational numbers is irrational (or simplifying gives an irrational number).