Vene T. Bogoslavov: Collection of Solved Problems #2

Solved Math Problems

Matematematika képletek

Simplify the following expression:

$5^{0} + 3^{- 2} \cdot {(\frac{1}{9})}^{- 1}$

Simplify the following expression:

${(a - b)}^{0} + 0, 1^{- 2} \cdot 10^{- 1} (a \neq b)$

Simplify the following expression:

${(\frac{1}{2})}^{- 2} - {(\frac{1}{4})}^{- 1} \cdot 2^{- 2}$

Simplify the following expression:

${({(1 \frac{1}{3})}^{- 1} - 2^{- 2})}^{- 3}$

Simplify the following expression:

$0, 1^{- 4} + 0, 01^{- 3} + 0, 001^{- 2} + 0, 0001^{- 1} + 0, 00001^{- 0}$

Simplify the following expression:

${(x + y)}^{0} + 0, 25^{- 1} + {(- 0, 5)}^{- 2} + {(0, 001^{0})}^{- 4} (x + y \neq 0)$

Simplify the following expression:

$2^{- 4} \cdot {(\frac{1}{4})}^{- 2}$

Simplify the following expression:

$2^{- 4} \cdot 0, 1^{- 4}$

Simplify the following expression:

${(\frac{5}{2} - {(\frac{3}{5})}^{- 1})}^{- 1}$

Simplify the following expression:

$\frac{2^{- 2} + 5 {(\frac{1}{2})}^{0}}{3 - {(\frac{2}{3})}^{- 2}}$

Simplify the following expression:

$\frac{3^{- 2} - {(\frac{3}{4})}^{- 2}}{2 - {(\frac{1}{5})}^{- 1}}$

Simplify the following expression:

${({(- \frac{2}{3})}^{- 3} + 3 \cdot 2^{- 3})}^{- 2}$

Simplify the following expression:

$0, 5^{- 1} + 0, 25^{- 2} + 0, 125^{- 3} + 0, 0625^{- 4}$

Simplify the following expression:

$1^{- 1} + 2^{- 2} + 3^{- 3} + {(- 1)}^{- 1} + {(- 2)}^{- 2} + {(- 3)}^{- 3}$

Simplify the following expression:

$\frac{{(\frac{1}{10})}^{- 3} \cdot 10^{5}}{{(\frac{1}{1000})}^{3} \cdot 1000^{- 5}}$

Simplify the following expression:

$\frac{0, 25^{- 2} \cdot 1, 75^{- 3}}{0, 75^{2} \cdot 1, 25^{3}}$

Simplify the following expression:

$\frac{0, 1^{- 2} - {(0, 4)}^{0}}{2 \frac{2}{3} {(\frac{2}{3})}^{- 3} + {(- \frac{1}{3})}^{- 1}}$

Simplify the following expression:

$A = \frac{1^{- 1} + 2^{- 2}}{{(\frac{2}{3})}^{- 2} + 4^{- 1} \cdot 5 + 0, 5^{- 2}}$

Simplify the following expression:

$A = \frac{{({(- 12)}^{- 3})}^{- 2} \cdot 75^{- 1} \cdot {(- 4)}^{0}}{{(25^{- 1})}^{4} \cdot 6^{6} \cdot 10^{4}}$

Hozza egyszerűbb alakra a következő kifejezést:

$\frac{27 x^{- 2} y^{- 3}}{3^{2} x^{- 4} y^{2}}$

Simplify the following expression:

$\frac{a^{- 3} a b^{- 2}}{2^{- 4} a^{3} b^{- 3}}$

Simplify the following expression:

$\frac{x^{- 5} + 2 x^{2} - x}{x^{- 3}}$

Simplify the following expression:

${(\frac{y^{- 2}}{x^{- 2}})}^{- 3} \div {(\frac{x^{- 1}}{y^{2}})}^{- 3}$

Simplify the following expression:

$(\frac{a^{- 3}}{b^{- 3}}) \div {(\frac{a^{- 2}}{b^{3}})}^{- 2}$

Simplify the following expression:

$(50^{x} + 30^{x} + 18^{x}) \cdot (5^{x} - 3^{x})$

Simplify the following expression:

$(25^{x} + 20^{x} + 16^{x}) \cdot (5^{x} - 4^{x})$

Simplify the following expression:

$\frac{a + b^{- 1}}{a + c {(b c + 1)}^{- 1}} - \frac{b^{- 1}}{a b c + a + c}$

Simplify the following expression:

$\frac{a b^{- 1} - a^{- 1} b}{a^{- 2} - 2 a^{- 1} b^{- 1} + b^{- 2}}$

Simplify the following expression:

${(\frac{2^{x} + 2^{- x}}{2})}^{2} - {(\frac{2^{x} - 2^{- x}}{2})}^{2}$

Simplify the following expression:

$\frac{x^{- 4} - 25 y^{- 2}}{x^{- 2} - 5 y^{- 1}} \cdot x^{2} y {(y + 5 x^{2})}^{- 1}$

Simplify the following expression:

$\frac{a^{- 4} - 9 b^{- 2}}{a^{- 2} - 3 b^{- 1}} \cdot \frac{{(b + 3 a^{2})}^{- 1}}{a^{- 2} b^{- 1}}$

Simplify the following expression:

$\frac{a^{- 2} + b^{- 2}}{a^{- 1} + b^{- 1}} \cdot {(\frac{a^{2} + b^{2}}{a b})}^{- 1}$

Simplify the following expression:

$({(\frac{5 x^{- 5}}{2 y^{- 2}})}^{- 2} \cdot {(\frac{y^{- 1}}{5 x^{- 1}})}^{- 3}) \div 10 x^{2} y^{- 3}$

Simplify the following expression:

$({(\frac{3 a^{2}}{4 b^{- 3}})}^{- 3} \div {(\frac{9 a^{- 2} b}{4})}^{- 2}) \cdot \frac{b^{7}}{12 a^{- 11}}$

Simplify the following expression:

$({(\frac{3 x^{- 3}}{5 y^{- 2}})}^{- 3} \div {(\frac{9 x^{- 2}}{5 y^{- 3}})}^{- 2}) \cdot \frac{x^{- 6} y}{15}$

Simplify the following expression:

$({(\frac{2 a^{- 2}}{3 a b^{- 3}})}^{- 4} \div {(\frac{4 a^{- 2}}{3 b^{- 3}})}^{- 3}) \cdot \frac{1}{12 a^{5} b^{- 2}}$

Simplify the following expression:

$a \sqrt{128}$

Rationalize the denominator of the fraction:

$\frac{9}{\sqrt{12}}$

Rationalize the denominator of the fraction:

$\frac{6}{\sqrt[3]{2}}$

Rationalize the denominator of the fraction:

$\frac{1}{2 + \sqrt{3}}$

Hozza egyszerűbb alakra a következő kifejezést:

$\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$

Calculate the following expression:

${(\sqrt{6 - 2 \sqrt{5}} - \sqrt{6 + 2 \sqrt{5}})}^{2}$

Simplify the following expression:

$\frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x y}} (\frac{\sqrt{y}}{x - \sqrt{x y}} + \frac{\sqrt{y}}{x + \sqrt{x y}})$

$x > 0; y > 0$

Simplify the following expression:

$\sqrt{6 - 4 \sqrt{2}}$

Simplify the following expression:

$(\sqrt[3]{1 + 2 \sqrt{6}} - \sqrt[6]{25 + 4 \sqrt{6}}) \cdot \sqrt[3]{2 \sqrt{6} - 1}$

Simplify the following expression:

$\frac{2 + i}{2 - i} + \frac{2 - i}{2 + i}$

Using the formula $x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ for solutions of a quadratic equation $a x^{2} + b x + c = 0$ , solve the equation:

$x^{2} - 5 x + 6 = 0$

Using the formula $x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ for solutions of a quadratic equation $a x^{2} + b x + c = 0$ , solve the equation:

$4 x^{2} - 17 x + 4 = 0$

Using the formula $x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ for solutions of a quadratic equation $a x^{2} + b x + c = 0$ , solve the equation:

$x^{2} - 9 \frac{1}{9} x + 1 = 0$

Using the formula $x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ for solutions of a quadratic equation $a x^{2} + b x + c = 0$ , solve the equation:

$x^{2} - 0, 11 x + 0, 001 = 0$

Using the formula $x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ for solutions of a quadratic equation $a x^{2} + b x + c = 0$ , solve the equation:

$x^{2} - 3 x - 28 = 0$

Using the formula $x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ for solutions of a quadratic equation $a x^{2} + b x + c = 0$ , solve the equation:

$2 x^{2} - 5 x - 25 = 0$

Solve the following equation:

$x^{2} - (3 - 2 \sqrt{2}) x + 4 - 3 \sqrt{2} = 0$

Solve the following equation:

$(1 + \sqrt{2}) x^{2} - 2 (1 - \sqrt{2}) x - 3 \sqrt{2} + 1 = 0$

Oldja meg az alábbi egyenletet:

$x^{2} - (3 + 2 \sqrt{5}) x + 7 + 3 \sqrt{5}$

MR-200 / 1069.a

Solve the exponential equation:

$2^{x - 3} = 16$

MR-202 / 1070.a

Solve the following exponential equation:

$16^{\frac{1}{x}} = 4^{\frac{x}{2}}$

MR-204 / 1071.b

Solve the following exponential equation:

$4^{x + 1} + 4^{x} = 320$

Solve the following exponential equation:

$2^{x - 1} - 2^{x - 3} = 3^{x - 2} - 3^{x - 3}$

MR-208 / 1076.a

$5^{x} - 5^{3 - x} = 20$

Prove the following identity::

$s i n^{4} α + c o s^{2} α + s i n^{2} α \cdot c o s^{2} α = 1$