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Chapter 5: Problem 30

Express using positive exponents and, if possible, simplify. $$\left(\frac{x}{3}\right)^{-4}$$

Short Answer

Expert verified
\(\left(\frac{x}{3}\right)^{-4} = \frac{81}{x^4}\)

Step by step solution

01

- Apply the Negative Exponent Rule

The rule for a negative exponent is: \ \(a^{-n} = \frac{1}{a^n}\). \ Apply this rule to the given expression: \ \(\left(\frac{x}{3}\right)^{-4} = \frac{1}{\left(\frac{x}{3}\right)^4}\).
02

- Simplify the Denominator

Simplify the denominator by applying the exponent to both the numerator and the denominator inside the fraction: \ \(\frac{1}{\left(\frac{x}{3}\right)^4} = \frac{1}{\left(\frac{x^4}{3^4}\right)}\).
03

- Invert and Simplify

Since \(\frac{1}{\left(\frac{x^4}{3^4}\right)}\) is the same as \(\frac{3^4}{x^4}\), invert the fraction: \ Hence, \(\left(\frac{x}{3}\right)^{-4} = \frac{3^4}{x^4}\).
04

- Compute the Exponents

Calculate the powers in the expression: \ \(3^4 = 81\) and \(x^4\) remains as it is. \ So, \(\frac{3^4}{x^4} = \frac{81}{x^4}\).
05

- Final Representation

Combine the calculated values to get the final simplified expression: \ \(\frac{81}{x^4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponent Rule
When you encounter a negative exponent, it might look intimidating, but it's actually quite simple to handle. The rule for a negative exponent is: \[a^{-n} = \frac{1}{a^n}\]This means you can turn the negative exponent into a positive one by taking the reciprocal of the base.
For example, \[\left(\frac{x}{3}\right)^{-4} = \frac{1}{\left(\frac{x}{3}\right)^4}\]This is just the first step in simplifying the expression. Don't skip this fundamental rule, as it lays the foundation for converting all negative exponents into positive ones!
Fractional Exponents
Next up is handling fractional exponents, which appear frequently in algebra. When you apply the exponent to a fraction, you raise both the numerator and the denominator to that exponent.
For the expression \[\left(\frac{x}{3}\right)^4\], we apply the exponent of 4 to both the numerator and the denominator separately:
\[\left(\frac{x}{3}\right)^4 = \frac{x^4}{3^4}\]After doing this, the fractions are easier to manage. Remember this: for a fraction \[(\frac{a}{b})^n\], just apply \[a^n\] and \[b^n\] independently to avoid mistakes.
Simplifying Expressions
Finally, let's combine all these steps and simplify the expression completely. After applying the negative exponent rule and handling fractional exponents, you should have something like \[\frac{1}{\left(\frac{x^4}{3^4}\right)}\].
Now, we know that dividing by a fraction is the same as multiplying by its reciprocal, so we can write: \[\frac{1}{\left(\frac{x^4}{3^4}\right)} = \frac{3^4}{x^4}\] Next, compute the powers: \[3^4 = 81\], so we get \[\frac{81}{x^4}\].
This is our simplified expression with positive exponents. Breaking down each step makes complex expressions much easier to handle!

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