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Chapter 1: Problem 24

Divide and simplify. $$ \frac{24 t^{9}}{8 t^{3}} $$

Short Answer

Expert verified
The simplified form is \( 3t^{6} \).

Step by step solution

01

Identify the Numerator and Denominator

The given fraction is \( \frac{24 t^{9}}{8 t^{3}} \). Here, the numerator is \( 24 t^{9} \) and the denominator is \( 8 t^{3} \).
02

Divide the Coefficients

Divide the coefficients of the numerator and the denominator: \( \frac{24}{8} = 3 \). So, the fraction becomes \( \frac{3 t^{9}}{t^{3}} \).
03

Apply the Law of Exponents

For the remaining variable part, apply the law of exponents which states \( \frac{a^m}{a^n} = a^{m-n} \). Here, we have \( \frac{t^{9}}{t^{3}} = t^{9-3} = t^{6} \).
04

Combine the Results

Combine the results of the coefficient and the variable part: \( 3 \times t^{6} = 3t^{6} \).

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Key Concepts

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

Numerator and Denominator
Understanding a fraction begins with identifying its two main parts: the numerator and the denominator. The fraction in this exercise is \( \frac{24 t^{9}}{8 t^{3}} \). Here, the numerator is the expression on the top, which is \( 24 t^{9} \), and the denominator is the expression on the bottom, \( 8 t^{3} \). The numerator tells us how many parts we have, while the denominator tells us into how many parts the whole is divided.
Recognizing these parts is crucial as we move forward in solving and simplifying the fraction.
Coefficients
Coefficients are the numerical parts of the terms in algebraic expressions. In the given fraction, the coefficients are \( 24 \) (from the numerator) and \( 8 \) (from the denominator).
To handle these, first separate them from the variables. Then, simply divide the coefficients just as you would with regular numbers:
  • \( \frac{24}{8} = 3 \)
Hence, the fraction reduces from \( \frac{24 t^{9}}{8 t^{3}} \) to \( \frac{3 t^{9}}{t^{3}} \).
Law of Exponents
The law of exponents is a rule that tells us how to handle powers in division. It states:
  • \( \frac{a^m}{a^n} = a^{m-n} \)

Applying this to the variable part of our fraction, we need to divide \( t^{9} \) by \( t^{3} \).
According to the law of exponents:
  • \( \frac{t^{9}}{t^{3}} = t^{9-3} = t^{6} \)
This means we subtract the exponent in the denominator from the exponent in the numerator.
Simplifying Fractions
Simplifying a fraction means reducing it to its simplest form. This involves dividing the coefficients and applying the law of exponents to the variables. We began with the fraction:
  • \( \frac{24 t^{9}}{8 t^{3}} \)

We divided the coefficients to get 3, and used the law of exponents to simplify \( \frac{t^{9}}{t^{3}} \) to \( t^{6} \).
Finally, putting it all together, we get the simplified result:
  • \( 3 t^{6} \)

This is the fraction in its simplest form! Simplification makes the expression easier to work with and understand.

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