What is the meaning of cdot and Idot
Answer: \(\cdot\) (written as \cdot) is the centered dot used for multiplication or the dot (scalar) product. “Idot” usually means \( \dot{I} \), i.e. I with a dot above it, which denotes the time derivative \( \dfrac{dI}{dt} \). Explanation: \(\cdot\): in arithmetic/algebra it denotes multiplication, e.g. \(3\cdot4=12\). In vector calculus it denotes the dot (scalar) […]
calculate (4 choose 2)
Answer: 6 Explanation: \(\binom{4}{2} = \frac{4!}{2!\,2!} = \frac{24}{2\cdot2} = 6.\) (The 6 pairs are 12, 13, 14, 23, 24, 34.)
25 times 4 equal
Answer: 100 Explanation: \(25 \times 4 = 100\). You can verify by adding: \(25 + 25 + 25 + 25 = 100\).
How to calculate matrix of technical coefficient
Answer: The technical-coefficient matrix \(A\) has entries \(a_{ij}=\dfrac{z_{ij}}{x_j}\). In matrix form \(A = Z \,\hat{x}^{-1}\), where \(Z\) is the matrix of interindustry flows and \(\hat{x}\) is the diagonal matrix with total outputs \(x_j\) on the diagonal. Explanation: Each \(a_{ij}\) is the amount of input from sector \(i\) required per unit of output of sector \(j\) […]
Solve Cos2x=cosx
Answer: \(x = 2\pi k\) or \(x = \frac{2\pi}{3} + 2\pi k\) or \(x = \frac{4\pi}{3} + 2\pi k\), for any integer \(k\). Explanation: Steps: Use the double-angle identity: \(\cos 2x = 2\cos^2 x – 1\). Set equal to \(\cos x\): \(2\cos^2 x – 1 = \cos x\). Let \(u=\cos x\). Solve the quadratic \(2u^2 […]
Write 0.6 as a fraction in simplest form.
Answer: \( \frac{3}{5} \) Explanation: \(0.6 = \frac{6}{10}\). Divide numerator and denominator by their GCD 2: \(\frac{6}{10} = \frac{3}{5}\). You can check \(3 \div 5 = 0.6\).
Which expression has a value of 20? 9 + (21 ÷ 3 + 2) × 5 9 plus left parenthesis 21 divided by 3 plus 2 right parenthesis times 5 9 + 21 ÷ (3 + 2 × 5) 9 plus 21 divided by left parenthesis 3 plus 2 times 5 right parenthesis (9 + 21) ÷ 3 + 2 × 5 left parenthesis 9 plus 21 right parenthesis divided by 3 plus 2 times 5 9 + 21 ÷ (3 + 2) × 5
Answer: (9 + 21) ÷ 3 + 2 × 5 Explanation: \((9+21)\div3 + 2\times5 = 30\div3 + 10 = 10 + 10 = 20.\)
(1,500,000 x 5/12) + (2,500,000 x 7/12) solve
Answer: \( \dfrac{6,250,000}{3}\approx 2,083,333.\overline{3} \) Explanation: \(1,500,000\times\frac{5}{12}=625,000\). \(2,500,000\times\frac{7}{12}=\frac{17,500,000}{12}=1,458,333.\overline{3}\). Sum = \(625,000+\frac{17,500,000}{12}=\frac{25,000,000}{12}=\frac{6,250,000}{3}\approx2,083,333.\overline{3}\).
Which expression is equal to 7/8 with Detailed Explaination
Answer: Several expressions equal to \(\tfrac{7}{8}\). Examples: \(\tfrac{14}{16},\ \tfrac{21}{24},\ \tfrac{35}{40},\ \tfrac{56}{64}\) \(0.875\) \(87.5\%\) \(1 – \tfrac{1}{8}\) \(\tfrac{3}{8} + \tfrac{4}{8}\) More generally, \(\dfrac{7k}{8k}\) for any nonzero integer \(k\). Explanation: Equivalent fractions: multiplying numerator and denominator of \(\tfrac{7}{8}\) by the same nonzero integer \(k\) gives \(\dfrac{7k}{8k}\). For example with \(k=2\) we get \(\tfrac{14}{16}\); with \(k=3\) we get […]
Factorize 4x-8 with other Examples
Answer: \(4x-8=4(x-2)\). Explanation: Factor out the greatest common factor (GCF) 4 from both terms: \(4x-8=4\cdot x-4\cdot 2=4(x-2)\). Check by distributing: \(4(x-2)=4x-8\). Other quick examples: \(6y+9=3(2y+3)\) \(12a^2b-8ab=4ab(3a-2)\) \(-5x+15=-5(x-3)\) \(9m^2-6m+3=3(3m^2-2m+1)\) \(10x^2-25x+5=5(2x^2-5x+1)\) Tip: always find the GCF of all terms, divide each term by it, and place the GCF outside parentheses.