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Chapter 8: Problem 7

Simplify completely. $$\frac{\frac{s^{3}}{r^{3}}}{\frac{s^{4}}{t}}$$

Short Answer

Expert verified
The simplified expression is: \[\frac{t}{r^3 \cdot s}\].

Step by step solution

01

Invert the second fraction and multiply

First, we rewrite the given expression as a multiplication by inverting the second fraction: \[\frac{\frac{s^3}{r^3}}{\frac{s^4}{t}} = \frac{s^3}{r^3} \cdot \frac{t}{s^4}\]
02

Apply exponent rules

Now, we will use the exponent rules to simplify the expression. Specifically, we will use the rule that states (a^m)^n = a^(m*n). As such: \[\frac{s^3}{r^3} \cdot \frac{t}{s^4} = \frac{s^3 \cdot t}{r^3 \cdot s^4}\]
03

Cancel out common factors

Next, we will cancel out any common factors in the numerator and denominator. We see that s^3 is a common factor: \[\frac{s^3 \cdot t}{r^3 \cdot s^4} = \frac{t}{r^3 \cdot s}\] Thus, the simplified expression is: \[\frac{t}{r^3 \cdot s}\]

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Key Concepts

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

Exponent Rules
Exponent rules are essential when dealing with algebraic expressions, especially ones involving fractions. The main rule used here is when multiplying powers with the same base, you subtract the exponents. Another useful rule is \(a^m \/ a^n = a^{m-n}\). In our exercise, we need to subtract the exponents of \(s\) because it appears both in the numerator and denominator.
By applying the rule, we transform \(s^3 \/ s^4\) into \(s^{3-4} = s^{-1}\). This simplification is pivotal as it lets us cancel out many components without altering the value of the expression. Understanding these rules will enable you to handle exponential terms with ease in any complex fraction.
Fraction Inversion
Fraction inversion is a vital technique in simplifying complex algebraic fractions. When dividing by a fraction, we multiply by its reciprocal. For instance, \(\frac{a}{b} \/ \frac{c}{d}\) becomes \(\frac{a}{b} \cdot \frac{d}{c}\).
This means flipping the numerator and denominator of the fraction being divided. In our example, we change \(\frac{s^4}{t}\) to \(\frac{t}{s^4}\), and multiply.
By using fraction inversion, we simplify the expression into a more manageable multiplication problem. This makes the next step of applying exponent rules much more straightforward.
Multiplication of Fractions
Multiplying fractions is a foundational skill in simplifying algebraic fractions. When you multiply two fractions, you multiply their numerators and their denominators separately. For example, \(\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\).
  • Combine the numerators: \(s^3 \cdot t\)
  • Combine the denominators: \(r^3 \cdot s^4\)
Once combined, it's crucial to look for common factors to simplify further.
Multiplying and then simplifying the expression helps in reducing it to its simplest form, ultimately providing a clearer result. In the exercise, this step allows the simplification of the fraction to \(\frac{t}{r^3 \cdot s}\), showcasing the combined power of multiplication and simplification techniques.

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Most popular questions from this chapter

Add or subtract, as indicated. $$\frac{2 y}{9 y^{2}-25}-\frac{2}{5-3 y}$$

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