answer:1. Given the expression ( 12345 times 6789 ), we need to express it in the form ( a times 10^p ), where ( 1 leq a < 10 ) and ( p ) is a positive integer. 2. First, break down each number ( 12345 ) and ( 6789 ) into a form involving powers of 10: [ 12345 = 1.2345 times 10^4 ] and [ 6789 = 6.789 times 10^3 ] 3. Substitute these expressions back into the original multiplication: [ 12345 times 6789 = (1.2345 times 10^4) times (6.789 times 10^3) ] 4. Combine the terms with powers of 10: [ = 1.2345 times 6.789 times 10^{4+3} = 1.2345 times 6.789 times 10^7 ] 5. Let ( a = 1.2345 times 6.789 ). Compute ( a ): [ a approx 1.2345 times 6.789 ] To estimate ( a ), we see that ( 1.2 times 6.8 approx 8.16 ). Thus, the refined value is still within ( 1 leq a < 10 ), confirming that the multiplication factor ( a ) falls in the required range. 6. Conclusion: Since ( 1.2345 times 6.789 approx 8.38 ) and given the form ( a times 10^p ), it concludes that ( p = 7 ). Thus, the value of ( p ) is: (boxed{7})

answer:The given inequality expands as [x^2 + y^2 + z^2 + 3 ge Cx + Cy + Cz.] Completing the square in (x), (y), and (z), we get: [ left( x - frac{C}{2} right)^2 + left( y - frac{C}{2} right)^2 + left( z - frac{C}{2} right)^2 + 3 - frac{3C^2}{4} ge 0. ] This inequality holds for all (x), (y), and (z) if and only if (3 - frac{3C^2}{4} ge 0), or (C^2 le 4). Thus, the largest possible value of (C) is (boxed{2}).

answer:To determine which differentiation operation is correct, we will analyze each option step by step: **Option A: left[ln left(2x+1right)right]'** Using the chain rule for differentiation, we have: [ left[ln left(2x+1right)right]' = dfrac{1}{2x+1} cdot dfrac{d}{dx}(2x+1) = dfrac{2}{2x+1} ] Since the result is dfrac{2}{2x+1}, option A is incorrect. **Option B: left(log _{2}xright)'** The derivative of log_{2}x with respect to x is: [ left(log _{2}xright)' = dfrac{1}{xln 2} ] This matches the given option B exactly, suggesting it might be correct. We will verify the other options before concluding. **Option C: left(3^{x}right)'** Using the derivative of an exponential function, we get: [ left(3^{x}right)' = 3^{x}ln 3 ] This matches the correction provided, indicating option C is incorrect. **Option D: left(x^{2}cos xright)'** Applying the product rule for differentiation, we have: [ left(x^{2}cos xright)' = 2xcos x + x^{2}(-sin x) = 2xcos x - x^{2}sin x ] Since the result is 2xcos x - x^{2}sin x, option D is incorrect. Given the analysis above, options A, C, and D are incorrect due to their respective derivations not matching the options provided. Option B is the only one that matches its derivation exactly. Therefore, the correct answer is boxed{text{B}}.

answer:Applying the vector triple product identity, we have for vectors mathbf{u}, mathbf{v}, mathbf{w}: [ mathbf{u} times (mathbf{v} times mathbf{w}) = (mathbf{u} cdot mathbf{w}) mathbf{v} - (mathbf{u} cdot mathbf{v}) mathbf{w}. ] So, applying it here: [ (mathbf{a} cdot mathbf{b}) mathbf{c} - (mathbf{a} cdot mathbf{c}) mathbf{b} + 2mathbf{d} = mathbf{0}. ] Assuming |mathbf{a}| = |mathbf{b}| = 1, we can simplify to: [ mathbf{c} - (mathbf{a} cdot mathbf{c}) mathbf{b} = -2mathbf{d}. ] Taking norms: [ |mathbf{c} - (mathbf{a} cdot mathbf{c}) mathbf{b}|^2 = | -2mathbf{d} |^2 = 8. ] Expanding left-hand side gives: [ |mathbf{c}|^2 - 2(mathbf{a} cdot mathbf{c}) + (mathbf{a} cdot mathbf{c})^2 |mathbf{b}|^2 = 9 - 6 (mathbf{a} cdot mathbf{c}) + (mathbf{a} cdot mathbf{c})^2 = 8. ] Thus, [ (mathbf{a} cdot mathbf{c})^2 - 6 (mathbf{a} cdot mathbf{c}) + 1 = 0 ] Using quadratic formula, [ (mathbf{a} cdot mathbf{c}) = frac{6 pm sqrt{36 - 4}}{2} = 3 pm sqrt{8}. ] Choosing the positive value to maximize mathbf{a} cdot mathbf{c}, get mathbf{a} cdot mathbf{c} = 3 + 2sqrt{2}. Since |mathbf{a}| = 1 and |mathbf{c}| = 3, [ costheta = frac{3 + 2sqrt{2}}{3}. ] Solving for theta gives theta approx 12.96^circ. Thus, the smallest angle possible is boxed{12.96^circ} indexing the choices available.