s each number rational or irrational?

Question

Select Rational or Irrational for each number.

Rational Irrational
0.15783456

3^√5

0.23¯¯¯¯

−√200

Answer 1

Let's evaluate each of the given numbers to determine if they are rational or irrational:

1. 0.15783456: This is a finite decimal, which can be expressed as a fraction (e.g., \( \frac{15783456}{100000000} \)). Therefore, it is Rational.

2. 3^√5: This is an expression involving an irrational number \( \sqrt{5} \). The base 3 is rational, but raising a rational number to an irrational exponent results in an irrational number. Therefore, \( 3^{\sqrt{5}} \) is Irrational.

3. 0.23¯¯¯¯: This notation represents a repeating decimal (0.23232323...). Such numbers can be expressed as a fraction. For example, \( 0.23̅ = \frac{23}{99} \). Therefore, it is Rational.

4. −√200: The square root of 200 can be simplified to \( -10\sqrt{2} \). Since \( \sqrt{2} \) is irrational, \( -10\sqrt{2} \) is also irrational. Therefore, it is Irrational.

In summary:

• 0.15783456: Rational
• 3^√5: Irrational
• 0.23¯¯¯¯: Rational
• −√200: Irrational