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Chapter 5: Problem 79

Use rules for exponents to simplify each expression. $$ \frac{(6 k)^{7}}{(6 k)^{4}} $$

Short Answer

Expert verified
The simplified expression is \(216k^3\).

Step by step solution

01

Apply the Quotient Rule for Exponents

The quotient rule for exponents states that when two exponential expressions with the same base are divided, you can subtract their exponents. Here, the base is \((6k)\), so subtract the exponents 7 and 4: \[ \frac{(6k)^7}{(6k)^4} = (6k)^{7-4} \] This simplifies to: \[ (6k)^3 \]
02

Expand the Simplified Expression

Now expand \((6k)^3\) by multiplying the base by itself three times:\[ (6k)^3 = 6k \times 6k \times 6k \] This will help in understanding what the expression composed of.
03

Simplify the Expanded Expression

Calculate the expanded expression:\[ 6 \times 6 \times 6 = 216 \] \[ k \times k \times k = k^3 \] Put these together: \[ (6k)^3 = 216k^3 \] This is the simplified form of the original expression.

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Key Concepts

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

Quotient Rule for Exponents
The quotient rule for exponents is a fundamental idea when working with expressions that involve powers. In simple terms, it helps us simplify expressions where we divide one exponential term by another, as long as they have the same base. The rule says that for any non-zero number \(a\) and integers \(m\) and \(n\), the expression \(\frac{a^m}{a^n}\) simplifies to \(a^{m-n}\). To break it down further:
  • Start by identifying the base, which is the same in both the numerator and the denominator. In our exercise, the base is \((6k)\).
  • Next, subtract the exponent in the denominator from the exponent in the numerator. For \((6k)^7\) and \((6k)^4\), subtract 4 from 7 to get 3.
Thus, using the quotient rule on \(\frac{(6k)^7}{(6k)^4}\), you end up with \((6k)^3\). Understanding this rule helps avoid common mistakes when simplifying more complex expressions.
Simplified Expression
Simplified expressions are expressions that have been reduced to their simplest form. This often means reducing the number of terms or simplifying the components as much as possible using mathematical rules. From our exercise, we used the quotient rule to simplify \(\frac{(6k)^7}{(6k)^4}\), which resulted in the expression \((6k)^3\). This is a shorter version compared to our original expression, and it's easier to work with. Why aim for simplification?
  • It helps in understanding the core of the expression.
  • It makes further calculations more manageable.
  • It can help identify patterns or relationships within the expression more clearly.
The simplified expression \((6k)^3\) is easy to interpret and provides a clearer picture of what the original expression represents.
Expanding Expressions
Expanding expressions involves multiplying the terms out to see all the components individually. It's a way to show the detailed breakdown of each element in an expression. For our simplified expression \((6k)^3\), expanding it involves multiplying the base \(6k\) by itself three times, as shown:\[(6k)^3 = 6k \times 6k \times 6k\]Step-by-step, here's what expanding does:
  • First, separate the numerical coefficient (6) from the variable \(k\).
  • Multiply the coefficient: \(6 \times 6 \times 6 = 216\).
  • Multiply the variable parts: \(k \times k \times k = k^3\).
Putting it all together gives you \(216k^3\). Expanding helps us understand the expression's full composition and makes it easier to perform other operations, such as adding or subtracting similar terms. It's essentially zooming in on the expression to see every detail.

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Most popular questions from this chapter

Is the product of a monomial and a monomial always a monomial? Explain.

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