Question

Solve $8 \mathrm{x}^{3 / 2 \mathrm{n}}-8 \mathrm{x}^{-3 / 2 \mathrm{n}}=63$

   Solve $8 \mathrm{x}^{3 / 2 \mathrm{n}}-8 \mathrm{x}^{-3 / 2 \mathrm{n}}=63$

Algebra and Trigonometry

Algebra and Trigonometry

StemEZ 1st Edition

Chapter 26, Problem 813

Instant Answer

Solved by Expert Nick Johnson

08/06/2021

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8y - 8/y = 63 Show more…

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Solve $8 \mathrm{x}^{3 / 2 \mathrm{n}}-8 \mathrm{x}^{-3 / 2 \mathrm{n}}=63$

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Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent and require unique strategies for solving them as compared to standard linear or polynomial equations. In these problems, understanding how to manipulate expressions that contain exponents is key, and this often involves rewriting terms using the properties of exponents to simplify the equation into a more tractable form.

Properties of Exponents

The properties of exponents are fundamental rules such as the product rule, quotient rule, and negative exponent rule. These allow one to combine or separate exponential expressions, simplify terms, and rewrite expressions into equivalent forms that can then be manipulated further. In the context of the equation at hand, understanding these properties is crucial when dealing with terms like x raised to a positive or negative fraction.

Substitution Method

The substitution method is a strategy used to simplify complex equations by replacing a repeated or complicated expression with a single variable. This technique transforms an exponential equation into a more familiar form, such as a quadratic equation, making it easier to solve. Once the substituted variable is solved, the original variable is recovered by reversing the substitution.

Quadratic Equations

Quadratic equations are second-degree polynomial equations that typically have two solutions. When an exponential equation is transformed using substitution, it may lead to a quadratic equation. Solving the quadratic equation using methods like factoring, completing the square, or the quadratic formula is a key step, with the subsequent back-substitution revealing the solution in the context of the original problem.

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