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Chapter 5: Problem 103

Simplify. Do not use negative exponents in the answer. \(\left(\frac{c^{3} d^{-4}}{c^{-1} d^{5}}\right)^{3}\)

Short Answer

Expert verified
\(\frac{c^{12}}{d^{27}}\)

Step by step solution

01

Apply Power of a Quotient Rule

When you have an expression of the form \( \left( \frac{a}{b} \right)^n \), you can apply the power to both the numerator and the denominator: \[ \left( \frac{c^{3} d^{-4}}{c^{-1} d^{5}} \right)^{3} = \frac{(c^{3} d^{-4})^{3}}{(c^{-1} d^{5})^{3}} \]
02

Apply Power of a Power Rule

Use the power of a power rule \((a^m)^n = a^{m \cdot n}\) to both the numerator and the denominator: \[ (c^{3})^{3} = c^{3 \times 3} = c^{9} \] \[ (d^{-4})^{3} = d^{-4 \times 3} = d^{-12} \] \[ (c^{-1})^{3} = c^{-1 \times 3} = c^{-3} \] \[ (d^{5})^{3} = d^{5 \times 3} = d^{15} \] Combine everything together: \[ \frac{c^{9} d^{-12}}{c^{-3} d^{15}} \]
03

Simplify Using Quotient of Powers Rule

Apply the quotient of powers rule \( \frac{a^m}{a^n} = a^{m-n} \): \[ \frac{c^{9}}{c^{-3}} = c^{9 - (-3)} = c^{12} \] \[ \frac{d^{-12}}{d^{15}} = d^{-12 - 15} = d^{-27} \] Combine: \[ c^{12} d^{-27} \]
04

Eliminate Negative Exponents

To eliminate negative exponents, express them as reciprocals: \[ c^{12} d^{-27} = c^{12} \cdot \frac{1}{d^{27}} \] This simplifies to: \[ \frac{c^{12}}{d^{27}} \]

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Key Concepts

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

Understanding the Power of a Quotient Rule
When you raise a fraction to a power, you typically need to apply the power to each part of the fraction. For example, if you have \( \left( \frac{a}{b} \right)^n \), you can distribute the power into the numerator and the denominator. This leads to the expression \( \frac{a^n}{b^n} \).

In the provided exercise, we start with \( \left( \frac{c^{3} d^{-4}}{c^{-1} d^{5}} \right)^{3} \). By applying the power of a quotient rule, each part inside the fraction is raised to the third power separately. This simplifies to \( \frac{(c^{3} d^{-4})^{3}}{(c^{-1} d^{5})^{3}} \), allowing further simplification of each term.
Mastering the Power of a Power Rule
One powerful tool in algebra simplification is the power of a power rule. Whenever you have an expression such as \( (a^m)^n \), you can simplify it by multiplying the exponents, resulting in \( a^{m \cdot n} \).

In our exercise, we apply this rule to each component of the fraction. Raising \( c^{3} \) to the third power becomes \( c^{9} \) and raising \( d^{-4} \) to the third power yields \( d^{-12} \). Similarly, raising \( c^{-1} \) to the third power gives \( c^{-3} \) and \( d^{5} \) raised to the third power gives \( d^{15} \). This process results in the expression \( \frac{c^{9} d^{-12}}{c^{-3} d^{15}} \) for further simplification.
Quotient of Powers Rule Explained
Simplification in algebra often involves the quotient of powers rule, aiding in reducing expressions in fractions. The rule states that \( \frac{a^m}{a^n} \) simplifies to \( a^{m-n} \). This concept helps manage terms that have the same base.

In our problem, applying this rule helps twofold:
  • For the \( c \) terms: \( \frac{c^{9}}{c^{-3}} = c^{9 - (-3)} = c^{12} \)
  • For the \( d \) terms: \( \frac{d^{-12}}{d^{15}} = d^{-12 - 15} = d^{-27} \)
This simplification gives us \( c^{12} d^{-27} \), preparing for the final step which involves negative exponents.
Handling Negative Exponents
Negative exponents can add complexity to algebra expressions, but understanding their basic concept helps tremendously. A negative exponent \( a^{-n} \) essentially means \( \frac{1}{a^n} \). The reciprocal function makes sense of negative exponents by flipping their base under a fraction.

Applying this to our earlier simplification \( c^{12} d^{-27} \), we need to manage the negative exponent in \( d^{-27} \). Transforming the term gives \( c^{12} \cdot \frac{1}{d^{27}} \), simplifying the expression further to \( \frac{c^{12}}{d^{27}} \). This step ensures the expression is clear and devoid of negative exponents, achieving the final answer as required by the exercise.

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