Simplify
(i) $\left(\frac{x^{a+b}}{x^{c}}\right)^{a-b}\left(\frac{x^{b+c}}{x^{a}}\right)^{b-c}\left(\frac{x^{c+a}}{x^{b}}\right)^{c-a}$
(ii) $\sqrt[m]{\frac{x^{1}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n]{\frac{x^{n}}{x^{1}}}$
(i) $\left(\frac{x^{a+b}}{x^{c}}\right)^{a-b}\left(\frac{x^{b+c}}{x^{a}}\right)^{b-c}\left(\frac{x^{c+a}}{x^{b}}\right)^{c-a}$
$\left(x^{a+b-c}\right)^{a-b}\left(x^{b+c-a}\right)^{b-c}\left(x^{c+a-b}\right)^{c-a}$
$\left(x^{a^{2}-b^{2}-c a+c b}\right)\left(x^{b^{2}-c^{2}-a b+a c}\right)\left(x^{c^{2}-a^{2}-b c+a b}\right)$
$x^{a^{2}-b^{2}-c a+c b+b^{2}-c^{2}-a b+a c+c^{2}-a^{2}-b c+a b}$
$x^{0}=1$
(ii) $\sqrt[1 m]{\frac{x^{1}}{x^{m}}} \times \sqrt[m n]{\frac{x^{m}}{x^{n}}} \times \sqrt[n 1]{\frac{x^{n}}{x^{1}}}$
$=\sqrt[\operatorname{lm}]{\mathrm{x}^{1}-\mathrm{m}} \times \sqrt[m n]{\mathrm{x}^{\mathrm{m}-\mathrm{n}}} \times \sqrt[n 1]{\mathrm{x}^{\mathrm{n}-1}}$
$=\left(x^{1-m}\right)^{\frac{1}{m}} \times\left(x^{m-n}\right)^{\frac{1}{m n}} \times\left(x^{n-1}\right)^{\frac{1}{n l}}$
$=(\mathrm{x})^{\frac{1-\mathrm{m}}{\mathrm{lm}}} \times(\mathrm{x})^{\frac{\mathrm{m}-\mathrm{n}}{\mathrm{mn}}} \times(\mathrm{x})^{\frac{\mathrm{n}-1}{\mathrm{nl}}}$
$=(\mathrm{x})^{\frac{1-\mathrm{m}}{\mathrm{lm}}+\frac{\mathrm{m}-\mathrm{n}}{\mathrm{mn}}+\frac{\mathrm{n}-1}{\mathrm{nl}}}$
$=(\mathrm{x})^{\mathrm{n}\left(\frac{1-\mathrm{m}}{\operatorname{lm}}\right)+\mathrm{l}\left(\frac{\mathrm{m}-\mathrm{n}}{\mathrm{mn}}\right)+\mathrm{m}\left(\frac{\mathrm{n}-\mathrm{l}}{\mathrm{nl}}\right)}$
$=(\mathrm{x})^{\frac{\mathrm{nl}-\mathrm{mn}+\mathrm{lm}-\mathrm{nl}+\mathrm{mn}-\mathrm{ml}}{\mathrm{mnl}}}$
$=(\mathrm{x})^{\frac{0}{\mathrm{mnl}}}$
$\mathrm{x}^{0}=1$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,