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How to Find the Power of a Quotient?

Published in Exponent Rules 3 mins read

Finding the power of a quotient involves applying the Power of a Quotient rule, which essentially distributes the exponent to both the numerator and the denominator.

Understanding the Power of a Quotient Rule

The Power of a Quotient rule states: The power of a quotient is equal to the power of each term in the numerator and denominator raised individually. In mathematical terms, this can be expressed as:

(a/b)n = an / bn

Where:

  • 'a' represents the numerator.
  • 'b' represents the denominator.
  • 'n' represents the exponent.

Essentially, when you have a fraction raised to a power, you raise both the top (numerator) and the bottom (denominator) to that same power.

Steps to Apply the Power of a Quotient Rule:

  1. Identify the Quotient: Recognize the fraction or quotient being raised to a power.
  2. Apply the Exponent to the Numerator: Raise the numerator to the given power.
  3. Apply the Exponent to the Denominator: Raise the denominator to the same given power.
  4. Simplify: If possible, simplify the resulting expression further.

Examples:

Example 1: Simple Numerical Quotient

Problem: Calculate (2/3)2

Solution:

  • Raise the numerator to the power: 22 = 4
  • Raise the denominator to the power: 32 = 9
  • Result: 4/9

So, (2/3)2 = 4/9.

Example 2: Algebraic Quotient

Problem: Calculate (x/y)3

Solution:

  • Raise the numerator to the power: x3
  • Raise the denominator to the power: y3
  • Result: x3 / y3

So, (x/y)3 = x3 / y3.

Example 3: Quotient with Coefficients and Variables

Problem: Calculate (2a/b)4

Solution:

  • Raise the numerator to the power: (2a)4 = 24 * a4 = 16a4
  • Raise the denominator to the power: b4
  • Result: 16a4 / b4

So, (2a/b)4 = 16a4 / b4

Table Summary of the Rule

Original Expression Applying Power of Quotient Result
(a/b)n an / bn an / bn
(2/3)2 22 / 32 4/9
(x/y)3 x3 / y3 x3 / y3
(2a/b)4 (2a)4 / b4 16a4 / b4

Practical Insights

  • This rule is incredibly useful for simplifying expressions involving fractions raised to a power.
  • Be careful to apply the exponent to all factors within the numerator and denominator.
  • Remember, if there are numbers within the fraction, you need to apply the power to them as well as the variables.
  • The Power of a Quotient rule can be combined with other exponent rules to solve more complex expressions.

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