Educational

Proof Prove: \[ \left(\frac{\sqrt{3\cdot 5^{-3}}}{\sqrt[3]{3^{-1}\cdot 5}}\right)\times \sqrt[6]{3\cdot 5^6} = \frac{3}{5} \] Proof \[ = \frac{(3\cdot 5^{-3})^{1/2}}{(3^{-1}\cdot 5)^{1/3}} \times (3\cdot 5^6)^{1/6} \] \[ = \frac{3^{1/2} \cdot 5^{-3/2}}{3^{-1/3}\cdot 5^{1/3}} \times 3^{1/6}\cdot 5 \] \[ = 3^{1/2 + 1/3 + 1/6} \cdot 5^{-3/2 – 1/3 + 1} \] \[ = 3^{\frac{3+2+1}{6}} \cdot 5^{\frac{-9-2+6}{6}} \] \[ = 3^1 \cdot […]

Simplify (√2/√3)^5 (6/7)^2 Simplify: \(\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^5 \left(\frac{6}{7}\right)^2\) Solution \[ \left(\frac{\sqrt{2}}{\sqrt{3}}\right)^5 \left(\frac{6}{7}\right)^2 \] \[ = \frac{(\sqrt{2})^5}{(\sqrt{3})^5} \cdot \frac{6^2}{7^2} \] \[ = \frac{2^{5/2}}{3^{5/2}} \cdot \frac{36}{49} \] \[ = \frac{4\sqrt{2}}{9\sqrt{3}} \cdot \frac{36}{49} \] \[ = \frac{144\sqrt{2}}{441\sqrt{3}} \] \[ = \frac{48\sqrt{2}}{147\sqrt{3}} \] \[ = \frac{48\sqrt{6}}{441} \] Final Answer: \[ \boxed{\frac{48\sqrt{6}}{441}} \] Next Question / Full Exercise

Simplify (x^-4 / y^-10)^(5/4) Simplify: \(\left(\frac{x^{-4}}{y^{-10}}\right)^{5/4}\) Solution \[ \left(\frac{x^{-4}}{y^{-10}}\right)^{5/4} \] \[ = (x^{-4} \cdot y^{10})^{5/4} \] \[ = x^{-4 \cdot \frac{5}{4}} \cdot y^{10 \cdot \frac{5}{4}} \] \[ = x^{-5} \cdot y^{25/2} \] \[ = \frac{y^{25/2}}{x^5} \] Final Answer: \[ \boxed{\frac{y^{25/2}}{x^5}} \] Next Question / Full Exercise

Simplify 5th root of 243x^10y^5z^10 Simplify: \(\sqrt[5]{243x^{10}y^{5}z^{10}}\) Solution \[ \sqrt[5]{243x^{10}y^{5}z^{10}} \] \[ = (243x^{10}y^{5}z^{10})^{1/5} \] \[ = 243^{1/5} \cdot x^{10/5} \cdot y^{5/5} \cdot z^{10/5} \] \[ = 3 \cdot x^2 \cdot y \cdot z^2 \] Final Answer: \[ \boxed{3x^2yz^2} \] Next Question / Full Exercise

Simplify √(x^3 y^-2) Simplify: \(\sqrt{x^3 y^{-2}}\) Solution \[ \sqrt{x^3 y^{-2}} \] \[ = (x^3 y^{-2})^{1/2} \] \[ = x^{3/2} \cdot y^{-1} \] \[ = \frac{x^{3/2}}{y} \] Final Answer: \[ \boxed{\frac{x^{3/2}}{y}} \] Next Question / Full Exercise

Simplify (√x^-3)^5 Simplify: \((\sqrt{x^{-3}})^5\) Solution \[ (\sqrt{x^{-3}})^5 \] \[ = \left((x^{-3})^{1/2}\right)^5 \] \[ = (x^{-3})^{5/2} \] \[ = x^{-15/2} \] \[ = \frac{1}{x^{15/2}} \] Final Answer: \[ \boxed{\frac{1}{x^{15/2}}} \] Next Question / Full Exercise

Find a, b, c and evaluate expression Given \(1176 = 2^a \cdot 3^b \cdot 7^c\), find \(a, b, c\) and evaluate \(2^a \cdot 3^b \cdot 7^{-c}\) Solution \[ 1176 = 2^3 \cdot 3^1 \cdot 7^2 \] \[ \Rightarrow a = 3,\; b = 1,\; c = 2 \] \[ 2^a \cdot 3^b \cdot 7^{-c} \]

Find x, y, z and evaluate expression Given \(2^x \cdot 3^y \cdot 5^z = 2160\), find \(x, y, z\) and evaluate \(2^x \cdot 2^{-y} \cdot 5^{-z}\) Solution \[ 2160 = 2^4 \cdot 3^3 \cdot 5^1 \] \[ \Rightarrow x = 4,\; y = 3,\; z = 1 \] \[ 2^x \cdot 2^{-y} \cdot 5^{-z} \]

Solve √(a/b) = (b/a)^(1-2x) Solve: \(\sqrt{\frac{a}{b}} = \left(\frac{b}{a}\right)^{1-2x}\) Solution \[ \sqrt{\frac{a}{b}} = \left(\frac{b}{a}\right)^{1-2x} \] \[ \Rightarrow \left(\frac{a}{b}\right)^{1/2} = \left(\frac{a}{b}\right)^{-(1-2x)} \] \[ \Rightarrow \left(\frac{a}{b}\right)^{1/2} = \left(\frac{a}{b}\right)^{2x-1} \] \[ \Rightarrow \frac{1}{2} = 2x – 1 \] \[ \Rightarrow 2x = \frac{3}{2} \] \[ \Rightarrow x = \frac{3}{4} \] Final Answer: \[ \boxed{x = \frac{3}{4}} \] Next Question

Solve 4^(x-1) × (0.5)^(3-2x) = (1/8)^x Solve: \(4^{x-1} \times (0.5)^{3-2x} = \left(\frac{1}{8}\right)^x\) Solution \[ 4^{x-1} \times (0.5)^{3-2x} = \left(\frac{1}{8}\right)^x \] \[ \Rightarrow (2^2)^{x-1} \times (2^{-1})^{3-2x} = (2^{-3})^x \] \[ \Rightarrow 2^{2x-2} \times 2^{-3+2x} = 2^{-3x} \] \[ \Rightarrow 2^{2x-2-3+2x} = 2^{-3x} \] \[ \Rightarrow 2^{4x-5} = 2^{-3x} \] \[ \Rightarrow 4x – 5 = -3x