This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — Here are the solutions to the questions: Question 1: The equivalent of sqrt(2^k) is: Step 1: Rewrite the square root as an exponent. The square root of a number can be expressed as raising that number to the power of (1)/(2). sqrt(x) = x^(1)/(2) So, for sqrt(2^k): sqrt(2^k) = (2^k)^(1)/(2) Step 2: Apply the power of a power rule. When raising a power to another power, multiply the exponents: (a^m)^n = a^mn. (2^k)^(1)/(2) = 2^k × (1)/(2) = 2^(k)/(2) Comparing this with the given options: A. 2^k B. 4^k C. 2^(k)/(2) D. sqrt(2^k) The correct option is C. C Question 2: Which of the following numbers is an irrational number? An irrational number is a number that cannot be expressed as a simple fraction (p)/(q), where p and q are integers and q ≠ 0. Its decimal representation is non-terminating and non-repeating. Let's examine each option: • A. (3)/(4): This is a fraction of two integers, so it is a rational number. • B. 0.333: This is a terminating decimal (or a repeating decimal if it's 0.333... = (1)/(3)). Both are rational numbers. • C. 33: This is an integer, which can be written as (33)/(1). It is a rational number. • D. sqrt(5): The square root of 5 is approximately 2.2360679.... This is a non-terminating and non-repeating decimal. It cannot be expressed as a simple fraction. Therefore, sqrt(5) is an irrational number. The correct option is D. D Question 3: A quarter of the volume of a cylinder with a radius of 14 mm and a height of 10 cm? ( = (22)/(7)) Step 1: Identify the given values and formula. Radius (r) = 14 mm Height (h) = 10 cm = (22)/(7) The formula for the volume of a cylinder is V = r^2 h. We need to find (1)/(4) of this volume. Step 2: Ensure consistent units. The radius is in millimeters and the height is in centimeters. Convert the radius to centimeters to match the height, as the options include cm^3. r = 14 mm = 1.4 cm h = 10 cm Step 3: Calculate the volume of the cylinder. V_cylinder = r^2 h V_cylinder = (22)/(7) × (1.4 cm)^2 × 10 cm V_cylinder = (22)/(7) × 1.96 cm^2 × 10 cm V_cylinder = 22 × (1.96)/(7) × 10 cm^3 V_cylinder = 22 × 0.28 × 10 cm^3 V_cylinder = 61.6 cm^3 Step 4: Calculate a quarter of the volume. V_quarter = (1)/(4) × V_cylinder V_quarter = (1)/(4) × 61.6 cm^3 V_quarter = 15.4 cm^3 Comparing this with the given options: A. 15.4 cm^3 B. 15.4 mm^3 C. 61.4 mm^3 D. 61.4 cm^3 The correct option is A. A What's next?