((5^(-1)times 7^(2))/(5^(2)times7^(-4)))^(7/2)times((5^(-2)times7^(3))/(5^(3)times7^(-5)))^(-5/2)

Jasonbradley

4 min read

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Jun 16, 2024

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Simplifying Complex Exponential Expressions

This article aims to break down the simplification process of the following complex exponential expression:

((5^(-1)times 7^(2))/(5^(2)times7^(-4)))^(7/2)times((5^(-2)times7^(3))/(5^(3)times7^(-5)))^(-5/2)

Let's approach this step-by-step:

1. Applying the Rules of Exponents

We'll utilize the following rules to simplify our expression:

• Product of Powers: x^m * xⁿ = x^(m+n)
• Quotient of Powers: x^m / xⁿ = x^(m-n)
• Power of a Power: (x^m)ⁿ = x^(m*n)

2. Simplifying the Inner Expressions

• First Expression: ((5^(-1)times 7^(2))/(5^(2)times7^(-4)))
  • Applying the Quotient of Powers rule: 5^(-1-2) * 7^(2-(-4)) = 5^-3 * 7⁶
• Second Expression: ((5^(-2)times7^(3))/(5^(3)times7^(-5)))
  • Applying the Quotient of Powers rule: 5^(-2-3) * 7^(3-(-5)) = 5^-5 * 7⁸

3. Applying the Power of a Power Rule

• First Expression: (5^-3 * 7⁶)^(7/2) = 5^(-3*(7/2)) * 7^(6*(7/2)) = 5^-21/2 * 7²¹
• Second Expression: (5^-5 * 7⁸)^(-5/2) = 5^(-5*(-5/2)) * 7^(8*(-5/2)) = 5^25/2 * 7^-20

4. Combining the Simplified Expressions

Now we have: (5^-21/2 * 7²¹) * (5^25/2 * 7^-20)

Applying the Product of Powers rule: 5^{(-21/2 + 25/2)} * 7^(21-20) = 5² * 7¹

5. Final Simplification

The simplified form of the original complex expression is: 5² * 7¹ = 25 * 7 = 175

Therefore, the value of the given complex exponential expression is 175.