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 4: To rationalize the denominator of the fraction (7)/(sqrt(10)-3), we multiply the fraction by: Step 1: Identify the conjugate of the denominator. The denominator is sqrt(10)-3. The conjugate of an expression of the form a-b is a+b. So, the conjugate of sqrt(10)-3 is sqrt(10)+3. Step 2: Determine the multiplying factor. To rationalize the denominator, we multiply the fraction by conjugateconjugate. In this case, we multiply by sqrt(10)+3sqrt(10)+3. The question asks what we multiply the fraction by, which refers to the expression used in the numerator and denominator to achieve rationalization. Among the given options, sqrt(10)+3 is the key part of this multiplying factor. The correct option is B. B Question 5: Given that a cuboid has dimensions of 3 m × 4 m × 5 m, its surface area must be equal to: Step 1: Identify the dimensions of the cuboid. Length (l) = 5 m Width (w) = 4 m Height (h) = 3 m Step 2: Use the formula for the surface area of a cuboid. The surface area (SA) of a cuboid is given by the formula: SA = 2(lw + lh + wh) Step 3: Substitute the dimensions into the formula and calculate. SA = 2((5 m)(4 m) + (5 m)(3 m) + (4 m)(3 m)) SA = 2(20 m^2 + 15 m^2 + 12 m^2) SA = 2(47 m^2) SA = 94 m^2 The correct option is B. B Question 6: The product of (8-sqrt(3)) and (8+sqrt(3)) is: Step 1: Recognize the pattern. This is a product of conjugates of the form (a-b)(a+b). Step 2: Apply the difference of squares formula. The formula for the difference of squares is (a-b)(a+b) = a^2 - b^2. Here, a=8 and b=sqrt(3). (8-sqrt(3))(8+sqrt(3)) = 8^2 - (sqrt(3))^2 Step 3: Calculate the result. 8^2 - (sqrt(3))^2 = 64 - 3 64 - 3 = 61 The correct option is A. A Question 7: Simplify 7sqrt(2) + 5sqrt(3) - 6sqrt(2) = Step 1: Identify like terms. Like terms are terms that have the same radical part. The terms with sqrt(2) are 7sqrt(2) and -6sqrt(2). The term with sqrt(3) is 5sqrt(3). Step 2: Combine the like terms. Combine the terms involving sqrt(2): 7sqrt(2) - 6sqrt(2) = (7-6)sqrt(2) = 1sqrt(2) = sqrt(2) The term 5sqrt(3) cannot be combined with sqrt(2) because they have different radical parts. Step 3: Write the simplified expression. sqrt(2) + 5sqrt(3) The correct option is C. C Got more? Send 'em!