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Chapter 6: Problem 394

Simplify. $$\left(\frac{2 j^{3}}{3 k}\right)^{4}$$

Short Answer

Expert verified
\( \frac{16j^{12}}{81k^4} \)

Step by step solution

01

Apply the Power to Both Numerator and Denominator

First, apply the power of 4 to both the numerator and the denominator inside the parenthesis: \( \left( \frac{2j^3}{3k} \right)^4 = \frac{(2j^3)^4}{(3k)^4} \)}
02

Simplify the Numerator

Next, raise each part of the numerator to the 4th power: \( (2j^3)^4 = 2^4 \cdot (j^3)^4 = 16j^{12} \).
03

Simplify the Denominator

Now, raise each part of the denominator to the 4th power: \( (3k)^4 = 3^4 \cdot (k)^4 = 81k^4 \).
04

Combine Results

Put the simplified numerator and denominator together to get the final expression: \( \frac{16j^{12}}{81k^4} \).

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Key Concepts

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

power of a fraction
When dealing with fractions raised to a power, it's crucial to apply the exponent to both the numerator and the denominator separately. This means if you have an expression \((\frac{a}{b})^n\), you must calculate \(a^n\) and \(b^n\) independently. In our exercise, we started with \(\left( \frac{2j^3}{3k} \right)^4\), and applied the exponent 4 to both the numerator \(2j^3\) and the denominator \(3k\). This crucial first step sets the foundation for further simplification.
exponent rules
Understanding and applying exponent rules is essential when simplifying expressions. Here are some key rules:
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\). This rule tells us to add the exponents when multiplying the same bases.

  • Power of a Power: \( (a^m)^n = a^{mn} \). This rule directs us to multiply the exponents when an exponent is raised to another power.

  • Power of a Product: \( (ab)^m = a^m \cdot b^m \). This rule aids us when an entire product is raised to a power. Apply the exponent to each factor separately.
Applying these rules, our example transforms the numerator \( (2j^3)^4 \) into \( 2^4 \cdot (j^3)^4 = 16j^{12} \) and the denominator \( (3k)^4 \) into \( 3^4 \cdot k^4 = 81k^4 \). Understanding these rules simplifies complex expressions efficiently.
simplifying expressions
Simplifying algebraic expressions involves reducing them to their most basic form, ensuring they're easier to interpret and solve. Here's how it's done in our example:
  • First, after applying the exponent to both the numerator and the denominator, we separated them into individual components for clarity.

  • Next, by raising each component to the 4th power, we simplified the expression step-by-step. We obtained \(2^4 = 16\) and \( (j^3)^4 = j^{12} \) in the numerator, and \(3^4 = 81\) and \((k)^4 = k^4 \) in the denominator.

  • Finally, combining these results gave us the simplified expression: \ \frac{16j^{12}}{81k^4} \.
Simplifying expressions helps in solving equations and understanding algebraic relationships in a more straightforward manner.

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Most popular questions from this chapter

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Simplify. (a) \(\frac{1}{q^{-10}}\) (b) \(\frac{1}{10^{-3}}\)

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