Simplify (4x^10)^20/(8x^20)^10

Question
Simplify (4x^10)^20/(8x^20)^10

Answer
STEP 1:   Observe the given term:

simplify this expression
The terms in the numerator and denominator are powers raised to some other power. For simplifying the given expression, you first need to simplify the numerator and denominator and then cancel any common factors.

Take the numerator. It can be simplified using the law:

                                         (anbm)p = anpbmp


Raise each factor to the power 20.

using the exponent law
STEP 2:    By following a similar method, you can get a simplified expression in the denominator also.

simplifying the expression

Since the term x200 is common to both the numerator and denominator, you can choose to cancel it out in the next step.

simplifying further

STEP 3:    Note that the base of the term in the numerator is 4, and that in the denominator is 8. Further simplification can be possible only if we make both the bases same.

Express the numerator and the denominator as powers of 2.

making the bases the same
STEP 4:    Again, simplify using the law of exponents,
          (am)n = amn.This gives:


simplify this algebraic fraction

STEP 5: Apply the quotient rule for exponents,

quotient rule

This gives:

final answer

The given expression thus simplifies to 210 or 1024.

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