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: Add: -3sqrt(11) + 4sqrt(11) Step 1: Identify like terms. Both terms, -3sqrt(11) and 4sqrt(11), have the same radical part, sqrt(11). This means they are like terms and can be combined. Step 2: Combine the coefficients. Add the numerical coefficients while keeping the radical part the same. -3sqrt(11) + 4sqrt(11) = (-3+4)sqrt(11) (-3+4)sqrt(11) = 1sqrt(11) 1sqrt(11) = sqrt(11) The simplified expression is sqrt(11). sqrt(11) Question 2: Multiply: 7sqrt(6) × (2)/(7sqrt(2)) Step 1: Rewrite the expression as a single fraction. 7sqrt(6) × (2)/(7sqrt(2)) = 7sqrt(6) × 27sqrt(2) Step 2: Simplify by canceling common factors. The 7 in the numerator and the 7 in the denominator cancel out. 7sqrt(6) × 27sqrt(2) = 2sqrt(6)sqrt(2) Step 3: Simplify the radical expression. Use the property sqrt(a)sqrt(b) = sqrt((a)/(b)). 2sqrt(6)sqrt(2) = 2sqrt((6)/(2)) 2sqrt((6)/(2)) = 2sqrt(3) The simplified expression is 2sqrt(3). 2sqrt(3) Question 3: Divide: ((1)/(2))^4 ÷ (1)/(8) Step 1: Calculate the value of ((1)/(2))^4. ((1)/(2))^4 = (1^4)/(2^4) = (1 × 1 × 1 × 1)/(2 × 2 × 2 × 2) = (1)/(16) Step 2: Rewrite the division as multiplication by the reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of (1)/(8) is (8)/(1) or 8. (1)/(16) ÷ (1)/(8) = (1)/(16) × (8)/(1) Step 3: Multiply the fractions. (1)/(16) × (8)/(1) = (1 × 8)/(16 × 1) = (8)/(16) Step 4: Simplify the resulting fraction. Divide both the numerator and the denominator by their greatest common divisor, which is 8. (8)/(16) = (8 ÷ 8)/(16 ÷ 8) = (1)/(2) The simplified result is (1)/(2). (1)/(2) Drop the next question.