Powers Of Fractions Formula

Powers of Fractions Formula:

\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]

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1. What is the Powers of Fractions Formula?

The Powers of Fractions Formula demonstrates that raising a fraction to a power is equivalent to raising both the numerator and denominator to that power separately. This fundamental algebraic identity is expressed as:

\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]

2. How Does the Formula Work?

The formula shows that:

\[ \left(\frac{a}{b}\right)^n = \frac{a \times a \times \cdots \times a}{b \times b \times \cdots \times b} = \frac{a^n}{b^n} \]

Where:

\( a \) — Numerator (any real number)
\( b \) — Denominator (any non-zero real number)
\( n \) — Power (any real number)

Explanation: When you raise a fraction to a power, you can distribute that power to both the numerator and denominator separately. This property holds true for all real numbers (except when b=0) and all exponents.

3. Practical Applications

Details: This formula is widely used in:

Simplifying algebraic expressions
Solving equations involving fractions
Scientific calculations with fractional units
Probability calculations
Engineering computations

4. Using the Calculator

Tips: Enter values for numerator (a), denominator (b ≠ 0), and power (n). The calculator will compute both sides of the equation to demonstrate their equality.

5. Frequently Asked Questions (FAQ)

Q1: Does this work for negative exponents?
A: Yes, the formula works for all real exponents, including negative ones. A negative exponent simply means taking the reciprocal.

Q2: What happens if the denominator is zero?
A: The formula is undefined when b=0, as division by zero is undefined in mathematics.

Q3: Can this be extended to more complex fractions?
A: Yes, the same principle applies to complex fractions and algebraic expressions in the numerator and denominator.

Q4: Does this work for fractional exponents?
A: Yes, the formula holds true for fractional (rational) exponents as well.

Q5: Why is this formula important?
A: It simplifies calculations and helps in manipulating algebraic expressions, making it a fundamental tool in mathematics.