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Chapter 7: Problem 50

Divide and, if possible, simplify. $$\frac{c}{d^{6}} \div \frac{c^{2}}{d}$$

Short Answer

Expert verified
\(\frac{1}{d^{5}c}\)

Step by step solution

01

Rewrite the Division as Multiplication

When dividing by a fraction, multiply by its reciprocal. Rewrite the expression from \(\frac{c}{d^{6}} \div \frac{c^{2}}{d}\) to \(\frac{c}{d^{6}} \times \frac{d}{c^{2}}\).
02

Multiply the Fractions

Multiply the numerators together and the denominators together. So, we have: \(\frac{c \times d}{d^{6} \times c^{2}} = \frac{cd}{d^{6}c^{2}}\).
03

Simplify the Fraction

Cancel any common factors in the numerator and the denominator. Here, cancel out the common factor of \(c\) and \(d\): \(\frac{cd}{d^{6}c^{2}} = \frac{1}{d^{5}c}\).

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Key Concepts

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

simplifying fractions
Simplifying fractions involves reducing a fraction to its simplest form. To do this, find the greatest common factor (GCF) of the numerator and the denominator and divide both by this number. For example, if you have \(\frac{8}{12}\), the GCF of 8 and 12 is 4. Dividing both the numerator and the denominator by 4, you get \(\frac{2}{3}\). This process makes fractions easier to work with and understand.
In our exercise, we simplified \(\frac{cd}{d^{6}c^{2}}\) to \(\frac{1}{d^{5}c}\) by canceling out common factors.
multiplying fractions
Multiplying fractions might sound complex but it's quite simple. You need to multiply the numerators together to get the new numerator, and the denominators together to get the new denominator. For example, \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\). In our problem, we transformed the division into multiplication by multiplying \(\frac{c}{d^6}\) by \(\frac{d}{c^2}\). This resulted in the fraction \(\frac{cd}{d^6c^2}\), which we then simplified.
reciprocal of a fraction
The reciprocal of a fraction is found by swapping its numerator and denominator. It's especially useful when dividing fractions. Instead of dividing by a fraction, you multiply by its reciprocal. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). In the original exercise, we converted the division \(\frac{c}{d^{6}} \div \frac{c^{2}}{d}\) into a multiplication problem by using the reciprocal, changing it to \(\frac{c}{d^{6}} \times \frac{d}{c^{2}}\).
common factors in fractions
Common factors are numbers that divide both the numerator and the denominator without leaving a remainder. Identifying and canceling common factors helps simplify fractions. In the exercise, we had \(\frac{cd}{d^{6}c^{2}}\). We canceled out common factors of c and d from the numerator and the denominator. This left us with \(\frac{1}{d^5c}\), a simplified fraction. Remember, simplifying fractions by cancelling common factors makes calculations easier and results clearer.

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