8th Grade Common Core Math Worksheets - Free Printable

To solve these problems, we need to understand the difference between rational and irrational numbers.

Step-by-Step Reasoning:

1. Rational Numbers: These are numbers that can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q$ is not zero. Their decimal forms either stop (terminate) or repeat a pattern forever.
* Examples: Integers (5, -3), fractions ($\frac{1}{2}$), terminating decimals (0.75), repeating decimals ($0.333...$).
* Square roots of perfect squares are rational (e.g., $\sqrt{4} = 2$, $\sqrt{9} = 3$).

2. Irrational Numbers: These are numbers that *cannot* be written as a simple fraction. Their decimal forms go on forever without repeating a pattern.
* Examples: $\pi$ (pi), $e$, and square roots of non-perfect squares (e.g., $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$).

Now let's solve each question:

1. Which number is rational?
* 1) $\pi$: Irrational (never-ending, non-repeating).
* 2) $\frac{5}{4}$: Rational (it is a fraction of integers).
* 3) $\sqrt{7}$: Irrational (7 is not a perfect square).
* 4) $\sqrt{\frac{3}{2}}$: Irrational (results in a non-repeating decimal).
* Answer: 2

2. Which is a rational number?
* 1) $\sqrt{8}$: Irrational (8 is not a perfect square).
* 2) $\pi$: Irrational.
* 3) $5\sqrt{9}$: Let's simplify. $\sqrt{9} = 3$. So, $5 \times 3 = 15$. 15 is an integer, which is rational.
* 4) $6\sqrt{2}$: Irrational (because $\sqrt{2}$ is irrational).
* Answer: 3

3. Which expression is rational?
* 1) $\pi$: Irrational.
* 2) $\sqrt{\frac{1}{2}}$: Irrational.
* 3) $\sqrt{3}$: Irrational.
* 4) $\sqrt{\frac{1}{4}}$: Simplify. $\sqrt{\frac{1}{4}} = \frac{1}{2}$. This is a fraction, so it is rational.
* Answer: 4

4. Which is an irrational number?
* 1) $\sqrt{9}$: Simplifies to 3. Rational.
* 2) 3.14: Terminating decimal. Rational. (Note: This is often used as an approximation for $\pi$, but as written, it is rational).
* 3) $\sqrt{3}$: 3 is not a perfect square. Irrational.
* 4) $\frac{3}{4}$: Fraction. Rational.
* Answer: 3

5. Which is an irrational number?
* 1) 0: Integer. Rational.
* 2) $\pi$: Irrational.
* 3) $-\frac{1}{3}$: Fraction. Rational.
* 4) $\sqrt{9}$: Simplifies to 3. Rational.
* Answer: 2

6. The number 0.14114111411114... is
* Look at the pattern: 14, 114, 1114, 11114. The number of 1s increases by one each time. Because the pattern changes and doesn't repeat a fixed block of digits, it never repeats.
* Therefore, it is irrational.
* Answer: 3

7. Which expression represents an irrational number?
* 1) $\sqrt{2}$: 2 is not a perfect square. Irrational.
* 2) $\frac{1}{2}$: Fraction. Rational.
* 3) 0.17: Terminating decimal. Rational.
* 4) 0: Integer. Rational.
* Answer: 1

8. Which number is irrational?
* 1) $\sqrt{9}$: Equals 3. Rational.
* 2) $\sqrt{8}$: 8 is not a perfect square. Irrational.
* 3) 0.3333: Terminating decimal. Rational. (If it were $0.\overline{3}$, it would be rational too, but as written, it's just a decimal).
* 4) $\frac{2}{3}$: Fraction. Rational.
* Answer: 2

9. Which is an irrational number?
* 1) $0.\overline{3}$: Repeating decimal. Rational.
* 2) $\frac{3}{8}$: Fraction. Rational.
* 3) $\sqrt{49}$: Equals 7. Rational.
* 4) $\pi$: Irrational.
* Answer: 4

10. Which number is irrational?
* 1) $\frac{5}{4}$: Fraction. Rational.
* 2) $0.\overline{3}$: Repeating decimal. Rational.
* 3) $\sqrt{121}$: Equals 11. Rational.
* 4) $\pi$: Irrational.
* Answer: 4

11. The value of $\sqrt{x^2 - 9}$ is a real and irrational number when $x$ is equal to
* We need the result inside the square root to be positive (for it to be real) and not a perfect square (for it to be irrational).
* Try 1) $x = 5$: $\sqrt{5^2 - 9} = \sqrt{25 - 9} = \sqrt{16} = 4$. (Rational).
* Try 2) $x = 0$: $\sqrt{0^2 - 9} = \sqrt{-9}$. (Not a real number).
* Try 3) $x = -3$: $\sqrt{(-3)^2 - 9} = \sqrt{9 - 9} = \sqrt{0} = 0$. (Rational).
* Try 4) $x = 4$: $\sqrt{4^2 - 9} = \sqrt{16 - 9} = \sqrt{7}$. $\sqrt{7}$ is irrational.
* Answer: 4

12. Which number below is irrational? $\sqrt{\frac{4}{9}}, \sqrt{20}, \sqrt{121}$
* $\sqrt{\frac{4}{9}} = \frac{2}{3}$ (Rational).
* $\sqrt{121} = 11$ (Rational).
* $\sqrt{20}$: 20 is not a perfect square (closest are 16 and 25). So $\sqrt{20}$ is irrational.
* Answer: $\sqrt{20}$

13. Given: $\frac{\sqrt{99}}{11}, \sqrt{164}, \sqrt{196}$. Identify the expression that is a rational number and explain why it is rational.
* $\sqrt{196}$: $14 \times 14 = 196$. So $\sqrt{196} = 14$. This is an integer, which is a rational number.
* $\sqrt{99}$ is not a perfect square ($9 \times 11$), so $\frac{\sqrt{99}}{11}$ is irrational.
* $\sqrt{164}$ is not a perfect square ($4 \times 41$), so it is irrational.
* Answer: $\sqrt{196}$ because it simplifies to the integer 14.

14. Write an irrational number and explain why it is irrational.
* Example: $\sqrt{2}$.
* Explanation: It cannot be written as a simple fraction, and its decimal form ($1.41421356...$) goes on forever without repeating.

Final Answer:
1) 2
2) 3
3) 4
4) 3
5) 2
6) 3
7) 1
8) 2
9) 4
10) 4
11) 4
12) $\sqrt{20}$
13) $\sqrt{196}$ (because it equals 14)
14) Example: $\sqrt{3}$ (because it is the square root of a non-perfect square and cannot be written as a fraction).