(TEST) Understanding Rational and Irrational Numbers

• 2

• 4

• 8

• 16

Answer explanation

The expression \(\sqrt{16}\) asks for the number that, when multiplied by itself, equals 16. The number 4 satisfies this condition, as \(4 \times 4 = 16\). Therefore, the correct answer is 4.

Which of the following numbers is irrational?

• 0.75

• 2

Answer explanation

The number \( \sqrt{3} \) is irrational because it cannot be expressed as a fraction of two integers. In contrast, \( \frac{1}{2} \), 0.75, and 2 are all rational numbers.

Tags

CCSS.8.NS.A.1

Answer explanation

To simplify (2^3) \cdot (2^4), use the property of exponents that states a^m \cdot a^n = a^{m+n}. Here, 3 + 4 = 7, so (2^3) \cdot (2^4) = 2^7. Thus, the correct answer is 2^7.

Tags

CCSS.8.EE.A.1

• Rational

• Irrational

Answer explanation

The number \(\frac{22}{7}\) is a fraction where both the numerator and denominator are integers. Since it can be expressed as a ratio of two integers, it is classified as a rational number.

Answer explanation

The expression x^{1/3} represents the cube root of x. In radical form, this is written as \sqrt[3]{x}, which is the correct choice. The other options represent different roots or powers.

Tags

CCSS.HSN.RN.A.2

• Because both numbers are irrational

• Because both numbers are rational

• Because the sum of any two numbers is irrational

Answer explanation

The sum of an irrational number, like \sqrt{2}, and a rational number, like 3, is always irrational. Therefore, the correct choice is that \sqrt{2} is irrational and 3 is rational.

Tags

CCSS.HSN.RN.B.3

Answer explanation

To simplify (x^3y^2)^2, apply the power of a power rule: multiply the exponents. This gives x^(3*2)y^(2*2) = x^6y^4. Thus, the correct answer is x^6y^4.

Tags

CCSS.HSA.APR.A.1

• Rational

• Irrational

Answer explanation

The product of a rational number, \(\frac{3}{4}\), and an irrational number, \(\sqrt{5}\), is always irrational. Therefore, \(\frac{3}{4} \cdot \sqrt{5}\) is irrational.

Tags

CCSS.HSN.RN.B.3

Answer explanation

To simplify \( \sqrt{50} \), we can factor it as \( \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \). Thus, the correct answer is \( 5\sqrt{2} \).

• Because 2 is irrational

• Because the product of a rational number and an irrational number is irrational

• Because both numbers are irrational

Answer explanation

The product of a rational number (2) and an irrational number (\sqrt{3}) is always irrational. Thus, the correct choice is that the product of a rational and an irrational number is irrational.

Tags

CCSS.HSN.RN.B.3