Verified Answer :

  • The constant 81 is rewritten as \(9^2\).
  • The power \(t^8\) is expressed as \((t^4)^2\).
  • The power \(u^{20}\) is expressed as \((u^{10})^2\).
  • Combining these, the expression is rewritten as \(\left(\frac{9 t^4}{u^{10}}\right)^2\).
  • Final Answer: \(\boxed{\left(\frac{9 t^4}{u^{10}}\right)^2}\)

Explanation

  1. Analysis of the Expression

The given expression is \(\frac{81 t^8}{u^{20}}\). To find an equivalent expression, we can use the properties of exponents, specifically \((x^a)^b = x^{ab}\) and \(\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n\). We look for ways to represent the constant \(81\) and the powers \(t^8\) and \(u^{20}\) as powers of other numbers.

  1. Rewriting as a Square

First, we examine the components of the numerator and the denominator. The constant 81 can be written as \(9^2\) because \(9 \times 9 = 81\). Alternatively, it can be written as \(3^4\) since \(3 \times 3 \times 3 \times 3 = 81\). The variable term \(t^8\) is equivalent to \((t^4)^2\), and the variable term \(u^{20}\) is equivalent to \((u^{10})^2\). Using these, we can rewrite the entire expression as a single fraction raised to the power of 2:
\[\frac{81 t^8}{u^{20}} = \frac{9^2 \cdot (t^4)^2}{(u^{10})^2} = \left(\frac{9 t^4}{u^{10}}\right)^2\]
This is a standard way to express the term as a perfect square.

  1. Alternative Equivalent Forms and Conclusion

We can also represent the expression using higher powers. Since \(81 = 3^4\), \(t^8 = (t^2)^4\), and \(u^{20} = (u^5)^4\), we can express the entire term as a fourth power:
\[\frac{81 t^8}{u^{20}} = \frac{3^4 \cdot (t^2)^4}{(u^5)^4} = \left(\frac{3 t^2}{u^5}\right)^4\]
Both \(\left(\frac{9 t^4}{u^{10}}\right)^2\) and \(\left(\frac{3 t^2}{u^5}\right)^4\) are valid equivalent expressions for the original term. We will present the squared form as the primary result.

Examples

In many engineering and physics problems, you often encounter complex formulas involving variables with high exponents. Simplifying these expressions is crucial for making them easier to handle in calculations. For instance, if you are calculating surface area or volume ratios, transforming an expression like \(\frac{81 t^8}{u^{20}}\) into \(\left(\frac{9 t^4}{u^{10}}\right)^2\) allows you to work with smaller base numbers and clearer relationships between variables. This technique is a fundamental step in algebra that helps maintain accuracy and clarity when solving advanced problems.

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