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Chapter 5: Problem 100

Simplify. Do not use negative exponents in the answer. \(\left(\frac{x^{4}}{3}\right)^{-4}\)

Short Answer

Expert verified
The simplified form is \( \frac{81}{x^{16}} \).

Step by step solution

01

Understanding Negative Exponents

When you have an expression with a negative exponent like \( (a/b)^{-n} \), it can be simplified by taking the reciprocal of the base and making the exponent positive: \( (a/b)^{-n} = (b/a)^{n} \). In this problem, the expression given is \( \left(\frac{x^{4}}{3}\right)^{-4} \).
02

Apply the Reciprocal Rule

Apply the rule for negative exponents to the expression \( \left(\frac{x^{4}}{3}\right)^{-4} \): This becomes \( \left(\frac{3}{x^{4}}\right)^{4} \).
03

Distribute the Exponent

Distribute the exponent of 4 to both the numerator and the denominator: \( \left(\frac{3}{x^{4}}\right)^{4} = \frac{3^{4}}{(x^{4})^{4}} \).
04

Calculate the Power of the Numerator

Calculate \( 3^{4} = 81 \). This will be the new numerator.
05

Simplify the Power of the Denominator

Apply the power to the power rule, \( (x^{m})^{n} = x^{m \cdot n} \), to the denominator: \((x^{4})^{4} = x^{16} \).
06

Write the Final Expression

Combine the calculated numerator and denominator: \( \frac{81}{x^{16}} \). This is the simplified form of the original expression without negative exponents.

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Key Concepts

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

Negative Exponents
Negative exponents might seem tricky at first, but they hold a simple rule.
When you encounter a negative exponent, it's a signal to take the reciprocal of the base and then apply the positive exponent. Here's the breakdown:
  • A negative exponent like \(a^{-n}\) means \(\frac{1}{a^n}\). You invert the base and make the exponent positive.
  • For a fraction \((\frac{a}{b})^{-n}\), convert it by swapping the numerator and denominator to make \((\frac{b}{a})^{n}\).

In our problem statement, \(\left(\frac{x^{4}}{3}\right)^{-4}\), a negative exponent \(-4\) flips the fraction to \(\left(\frac{3}{x^{4}}\right)^{4}\). This equals magnifying the reciprocal's effect by four times.
Power of a Power
When dealing with exponents, there's a useful rule known as the "power of a power" rule. This is especially helpful when an exponent is raised to another exponent.
The rule states: \((x^{m})^{n} = x^{m \cdot n}\).
  • If you see something like \((x^{2})^{3}\), apply the rule to get: \(x^{2 \cdot 3} = x^{6}\).

In our earlier expression \(\left(\frac{3}{x^{4}}\right)^{4}\), you need to distribute the exponent 4 across both the parts of the fraction.
  • The power applies separately to the numerator, 3, resulting in \(3^{4}\), which calculates to 81.
  • And to the denominator, \((x^{4})^{4}\), applying the rule gives \(x^{16}\).
This results in the simplified expression \(\frac{81}{x^{16}}\).
Reciprocal of a Fraction
The reciprocal of a fraction involves flipping the numerator and the denominator.
We use this concept when working with negative exponents, where we convert them to positive exponents by taking the reciprocal.
  • For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
  • When applied to an expression with a negative exponent, like \((\frac{a}{b})^{-n}\), this becomes \((\frac{b}{a})^{n}\).

In the given exercise \(\left(\frac{x^{4}}{3}\right)^{-4}\), taking the reciprocal first step changes it into \(\left(\frac{3}{x^{4}}\right)^{4}\). Now, the fraction is ready for further simplification and exponent application. This step ensures your solution is free from negative exponents, yielding a cleaner final result.

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Most popular questions from this chapter

Explain the error: \((x+3)(x-2)=x^{2}-6\)

Perform the indicated operations to simplify each expression, if possible. a. \(3 a+(4 a-1)+(6 a+2)\) b. \(3 a(4 a-1)(6 a+2)\)

Perform the indicated operations to simplify each expression, if possible. $$ \text { a. }(a+3)+\left(a^{2}-3 a+9\right) \text { b. }(a+3)\left(a^{2}-3 a+9\right) $$

Perform each division. $$ \frac{3 b^{2}-5 b+2}{3 b-2} $$

Perform the indicated operations. a. \(\frac{16 x^{2}-16 x-5}{4 x}\) b. \(\frac{16 x^{2}-16 x-5}{4 x+1}\)
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