Simplifying Exponential and Rational Expressions

Fundamental Rules of Exponents

To simplify rational expressions containing powers and variables, several core algebraic exponent laws must be applied:

  • Power of a Quotient Rule:(ab)n=anbn\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}

  • Power of a Product Rule:(a⋅b)n=an⋅bn(a \cdot b)^n = a^n \cdot b^n

  • Power of a Power Rule:(am)n=am⋅n(a^m)^n = a^{m \cdot n}

  • Quotient Rule for Exponents:aman=am−n\frac{a^m}{a^n} = a^{m - n}

  • Negative Exponent Rule:a−n=1ana^{-n} = \frac{1}{a^n}

Problem Statement

Simplify the following algebraic expression completely, leaving all exponents as positive values:

Algebraic expression with exponents

(2x95y4)5⋅(53y622x−5)\left( \frac{2x^9}{5y^4} \right)^5 \cdot \left( \frac{5^3 y^6}{2^2 x^{-5}} \right)

Step-by-Step Algebraic Simplification

  • Step 1: Expand the outer exponent in the first fraction

    • Apply the Power of a Quotient Rule and Power of a Product Rule to the term (2x95y4)5\left( \frac{2x^9}{5y^4} \right)^5:
    • For the numerator:     (2x9)5=25⋅(x9)5=25x9×5=25x45(2x^9)^5 = 2^5 \cdot (x^9)^5 = 2^5 x^{9 \times 5} = 2^5 x^{45}
    • For the denominator:     (5y4)5=55⋅(y4)5=55y4×5=55y20(5y^4)^5 = 5^5 \cdot (y^4)^5 = 5^5 y^{4 \times 5} = 5^5 y^{20}
    • Resulting fraction for the first term:     25x4555y20\frac{2^5 x^{45}}{5^5 y^{20}}

  • Step 2: Combine the two factors into a single rational expression

    • Multiply the expanded first term by the second term (53y622x−5)\left( \frac{5^3 y^6}{2^2 x^{-5}} \right)25x4555y20⋅53y622x−5=25⋅53⋅x45⋅y622⋅55⋅x−5⋅y20\frac{2^5 x^{45}}{5^5 y^{20}} \cdot \frac{5^3 y^6}{2^2 x^{-5}} = \frac{2^5 \cdot 5^3 \cdot x^{45} \cdot y^6}{2^2 \cdot 5^5 \cdot x^{-5} \cdot y^{20}}

  • Step 3: Simplify each base component using the Quotient Rule for Exponents

    • Base 22:2522=25−2=23=8\frac{2^5}{2^2} = 2^{5 - 2} = 2^3 = 8
    • Base 55:5355=53−5=5−2=152=125\frac{5^3}{5^5} = 5^{3 - 5} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25}
    • Base xx:x45x−5=x45−(−5)=x45+5=x50\frac{x^{45}}{x^{-5}} = x^{45 - (-5)} = x^{45 + 5} = x^{50}
    • Base yy:y6y20=y6−20=y−14=1y14\frac{y^6}{y^{20}} = y^{6 - 20} = y^{-14} = \frac{1}{y^{14}}

  • Step 4: Combine all simplified bases into the final fraction

    • Place terms with positive exponents in the numerator and terms with negative exponent denominators into the denominator:     23⋅x5052⋅y14=8x5025y14\frac{2^3 \cdot x^{50}}{5^2 \cdot y^{14}} = \frac{8x^{50}}{25y^{14}}

Summary of Intermediate and Final Results

  • Original Expression:(2x95y4)5⋅(53y622x−5)\left( \frac{2x^9}{5y^4} \right)^5 \cdot \left( \frac{5^3 y^6}{2^2 x^{-5}} \right)

  • Fully Expanded Expression:25⋅53⋅x45⋅y622⋅55⋅x−5⋅y20\frac{2^5 \cdot 5^3 \cdot x^{45} \cdot y^6}{2^2 \cdot 5^5 \cdot x^{-5} \cdot y^{20}}

  • Final Simplified Answer:8x5025y14\frac{8x^{50}}{25y^{14}}