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. (a) √10 + 27 (b) 12 + 4/5 (c) 21 × √3 (d) 12/23 × 5/11 each has columns for result is rational, result is irrational, and reason with a select dropdown.

Explanation:

Part (a)

Step 1: Identify Number Types

$\sqrt{10}$ is irrational (since 10 is not a perfect square, its square root is non - repeating, non - terminating decimal), and 27 is rational (it can be written as $\frac{27}{1}$).

Step 2: Analyze Sum of Rational and Irrational

The sum of a rational number and an irrational number is always irrational. So, $\sqrt{10}+27$ is irrational. The reason is: The sum of a rational number and an irrational number is irrational.

Part (b)

Step 1: Identify Number Types

12 is rational (can be written as $\frac{12}{1}$) and $\frac{4}{5}$ is rational (it is a fraction of two integers with a non - zero denominator).

Step 2: Analyze Sum of Rationals

The sum of two rational numbers is rational. We calculate $12+\frac{4}{5}=\frac{12\times5 + 4}{5}=\frac{60 + 4}{5}=\frac{64}{5}$, which is a rational number. The reason is: The sum of two rational numbers is rational.

Part (c)

Step 1: Identify Number Types

21 is rational (can be written as $\frac{21}{1}$) and $\sqrt{3}$ is irrational (since 3 is not a perfect square, its square root is non - repeating, non - terminating decimal).

Step 2: Analyze Product of Rational and Irrational

The product of a non - zero rational number and an irrational number is irrational. So, $21\times\sqrt{3}$ is irrational. The reason is: The product of a non - zero rational number and an irrational number is irrational.

Part (d)

Step 1: Identify Number Types

$\frac{12}{23}$ and $\frac{5}{11}$ are both rational (they are fractions of two integers with non - zero denominators).

Step 2: Analyze Product of Rationals

The product of two rational numbers is rational. We calculate $\frac{12}{23}\times\frac{5}{11}=\frac{12\times5}{23\times11}=\frac{60}{253}$, which is a rational number. The reason is: The product of two rational numbers is rational.

Answer:

s:
(a) Result is Irrational. Reason: The sum of a rational number and an irrational number is irrational.
(b) Result is Rational. Reason: The sum of two rational numbers is rational.
(c) Result is Irrational. Reason: The product of a non - zero rational number and an irrational number is irrational.
(d) Result is Rational. Reason: The product of two rational numbers is rational.