Mathematics Questions and Answers

\(\frac{1}{\sqrt{3}+2}=\frac{1}{\sqrt{3}+2}\times \frac{\sqrt{3}-2}{\sqrt{3}-2}\\=\frac{\sqrt{3}-2}{(\sqrt{3})^{2} -2^{2}}\\=\frac{\sqrt{3}-2}{3-4}=\frac{\sqrt{3}-2}{1}\\=-\sqrt{3}+2\\=2-\sqrt{3}\)

\(\frac{\frac{1}{10}\times\frac{2}{3}+\frac{1}{4}}{\frac{\frac{1}{2}}{\frac{3}{5}}-\frac{1}{4}}\\Numerator \hspace{1mm}\frac{1}{10}\times\frac{2}{3}+\frac{1}{4} = \frac{1}{5}+\frac{1}{4}\\=\frac{4+15}{60}=\frac{19}{60}\\denominator\hspace{1mm}= \frac{\frac{1}{2}}{\frac{3}{5}}-\frac{1}{4}=\frac{1}{2}\times\frac{5}{3}-\frac{1}{4}\\=\frac{5}{6}-\frac{1}{4}\\=\frac{10-3}{12}\\=\frac{7}{12}\\\frac{Numerator}{denominator}=\frac{\frac{19}{60}}{\frac{7}{12}}\\=\frac{19}{60}\times\frac{12}{7}=\frac{19}{35}\)

³√4^2x = 16
this implies that (³√4^2x)³ = (16)³
4^2x = 4^2*3
4^2x = 4⁶
∴ 2x = 6
x = 3

451₆ - P₇ = 305₆
P₇ = 451₆ - 305₆
P₇ = 142₆
convert 142₆ = 1 * 6² + 4 * 6¹ + 2 * 6⁰
= 36 + 24 + 2
= 62
Convert 62₁₀ to base 7
62/7 = 8 R 6
8/7 = 1 R 1
1/7 = 0 R 1
∴P₇ = 116₇

6log_x2 - 3log_x3 = 3log₅0.2
= logx²⁶ - 3logx³³ = log5^(0.2)3
= logx(64/27) = log5(1/5)³
logx(64/27) = log5(1/125)
let logx(64/27) = y
∴x^y = 64/27
and log5(1/125) = y
∴ 5^y = 1/125
5^y = 125^-1
5^y = 5^-3
∴ y = -3
substitute y = -3 in x^y = 64/27
implies x^-3 = 64/27
1/x³ = 64/27
64x³ = 27
x³ = 27/64
x³ = ³√27/64
x = 3/4