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Chapter 7: Problem 14

Multiply and, if possible, simplify. $$\frac{10}{t^{7}} \cdot \frac{3 t^{2}}{25 t}$$

Short Answer

Expert verified
\(\frac{6}{5t^6}\)

Step by step solution

01

- Multiply the Numerators

Multiply the numerators of the fractions: \ \(10 \times 3t^2 = 30t^2\)
02

- Multiply the Denominators

Multiply the denominators of the fractions: \ \(t^7 \times 25t = 25t^8\)
03

- Form the Combined Fraction

Combine the results from Step 1 and Step 2 into a single fraction: \(\frac{30t^2}{25t^8}\)
04

- Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by the greatest common divisor. In this case, divide by \25t^2\: \(\frac{30}{25} \frac{t^2}{t^8} = \frac{6}{5} \frac{1}{t^6} = \frac{6}{5t^6}\)

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Key Concepts

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

Numerator
In any fraction, the numerator is the top part. It tells us how many parts of the whole we have. In the fraction \(\frac{10}{t^{7}} \), 10 is the numerator because it is on top. When multiplying fractions, you multiply the numerators together. For example, in this problem, we multiply the numerators 10 and 3t^2 to get 30t^2.
Denominator
The denominator is the bottom part of the fraction. It shows the total number of equal parts into which the whole is divided. In the fraction \(\frac{10}{t^{7}} \), the denominator is t^7. To multiply fractions, you also multiply the denominators together. In this exercise, we multiply t^7 and 25t to get 25t^8.
Simplifying Fractions
After multiplying the numerators and denominators, you may get a large fraction. To make it simpler, we simplify the fraction by finding and dividing by common factors. In \(\frac{30t^2}{25t^8}\), we notice both 30 and 25 share a common factor of 5. Dividing them by 5, and simplifying the term involving t, we simplify the fraction to \(\frac{6}{5t^6}\). Simplifying makes the fraction easier to understand and use.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. When simplifying fractions, finding the GCD helps us reduce the fraction to its simplest form. For example, the GCD of 30 and 25 is 5. By dividing both the numerator and the denominator by their GCD, we simplify \(\frac{30}{25}\) to \(\frac{6}{5}\), making the final simplified fraction \(\frac{6}{5t^6}\).

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