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Simplify the following expression. Write the final answer in exponential form when necessary. Assume that all variables represent non-negative values.



$\frac{\left(x^{\frac{4}{3}}\right)^{\frac{9}{8}}}{\left(x^{\frac{4}{3}}\right)^{\frac{9}{8}}}=\square$



(Simplify your answer. Use integers or fractions for any numbers in the expression. Type your answer using exponential notation.)

Verified Answer :

  • Recognize the numerator and denominator are identical.
  • Simplify the exponents using the rule \((x^a)^b = x^{ab}\) to get \(\frac{x^{3/2}}{x^{3/2}}\).
  • Simplify the fraction to 1, which can be written in exponential form as \(x^0\).
  • The final simplified expression is \(\boxed{x^0}\).

Explanation

  1. Analyze the expression

We are given the expression \(\frac{\left(x^{\frac{4}{3}}\right)^{\frac{9}{8}}}{\left(x^{\frac{4}{3}}\right)^{\frac{9}{8}}}\) and need to simplify it. First, observe that the numerator and the denominator are identical. Any non-zero expression divided by itself is 1. We assume \(x eq 0\) to ensure the expression is defined and non-zero.

  1. Simplify the expression and write in exponential form

Applying the exponent rule \((x^a)^b = x^{ab},\) we can simplify the numerator and the denominator:
\[\left(x^{\frac{4}{3}}\right)^{\frac{9}{8}} = x^{\frac{4}{3} \times \frac{9}{8}} = x^{\frac{36}{24}} = x^{\frac{3}{2}}.\]
Thus, the expression becomes \(\frac{x^{\frac{3}{2}}}{x^{\frac{3}{2}}}.\) For \(x eq 0,\) this simplifies to \(1.\) Since we need to express the result in exponential form, we can write \(1\) as \(x^0.\)

Examples

In scientific computing and data analysis, identifying expressions that simplify to identity (1) is a common task. For instance, if you are working with normalization factors in a simulation where you divide a variable by its own scale, recognizing that the expression simplifies to 1 can significantly reduce the computational load and help verify the consistency of your mathematical models. This applies to various fields, including physics, engineering, and finance, where ratio-based metrics are essential.

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