Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Simplifying Algebraic Fractions
Below are a series of algebraic fractions that can be simplified. Simplify each fraction by applying the properties of exponents and the rules of fractions.
6. \(\frac{x^2}{x}\)
- Simplify by subtracting the exponents of \(x\).
7. \(\frac{z^4}{z^5}\)
- Simplify by subtracting the exponents of \(z\).
8. \(\frac{a^2b^2}{ab^2}\)
- Simplify by dividing the numerator by the denominator, canceling out the common terms.
9. \(\frac{d^5 f^3}{df}\)
- Simplify by dividing the numerator by the denominator, canceling out the common terms.
10. \(\frac{mn^2}{mn^2}\)
- Simplify by canceling out the common terms in the numerator and the denominator.
Detailed solutions:
#### 6. Simplification of \(\frac{x^2}{x}\)
\[
\frac{x^2}{x} = \frac{x \cdot x}{x} = x^{2-1} = x
\]
#### 7. Simplification of \(\frac{z^4}{z^5}\)
\[
\frac{z^4}{z^5} = \frac{z \cdot z \cdot z \cdot z}{z \cdot z \cdot z \cdot z \cdot z} = z^{4-5} = z^{-1} = \frac{1}{z}
\]
#### 8. Simplification of \(\frac{a^2b^2}{ab^2}\)
\[
\frac{a^2 b^2}{a b^2} = \frac{a \cdot a \cdot b \cdot b}{a \cdot b \cdot b} = a^{2-1} \cdot b^{2-2} = a \cdot 1 = a
\]
#### 9. Simplification of \(\frac{d^5 f^3}{df}\)
\[
\frac{d^5 f^3}{d f} = \frac{d \cdot d \cdot d \cdot d \cdot](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ede44e8-0d35-4c10-b39d-8c57b9fbf925%2F0b776f20-0693-4a9f-8d9d-fc85c4125030%2Flmwv9n8_processed.png&w=3840&q=75)
Transcribed Image Text:### Simplifying Algebraic Fractions
Below are a series of algebraic fractions that can be simplified. Simplify each fraction by applying the properties of exponents and the rules of fractions.
6. \(\frac{x^2}{x}\)
- Simplify by subtracting the exponents of \(x\).
7. \(\frac{z^4}{z^5}\)
- Simplify by subtracting the exponents of \(z\).
8. \(\frac{a^2b^2}{ab^2}\)
- Simplify by dividing the numerator by the denominator, canceling out the common terms.
9. \(\frac{d^5 f^3}{df}\)
- Simplify by dividing the numerator by the denominator, canceling out the common terms.
10. \(\frac{mn^2}{mn^2}\)
- Simplify by canceling out the common terms in the numerator and the denominator.
Detailed solutions:
#### 6. Simplification of \(\frac{x^2}{x}\)
\[
\frac{x^2}{x} = \frac{x \cdot x}{x} = x^{2-1} = x
\]
#### 7. Simplification of \(\frac{z^4}{z^5}\)
\[
\frac{z^4}{z^5} = \frac{z \cdot z \cdot z \cdot z}{z \cdot z \cdot z \cdot z \cdot z} = z^{4-5} = z^{-1} = \frac{1}{z}
\]
#### 8. Simplification of \(\frac{a^2b^2}{ab^2}\)
\[
\frac{a^2 b^2}{a b^2} = \frac{a \cdot a \cdot b \cdot b}{a \cdot b \cdot b} = a^{2-1} \cdot b^{2-2} = a \cdot 1 = a
\]
#### 9. Simplification of \(\frac{d^5 f^3}{df}\)
\[
\frac{d^5 f^3}{d f} = \frac{d \cdot d \cdot d \cdot d \cdot
Transcribed Image Text:### Simplifying Algebraic Expressions
#### Directions: Simplify the following expressions.
1. \(\dfrac{r^6}{r}\)
2. \(\dfrac{s^3 t^4}{s^2 t^2}\)
3. \(\dfrac{q^3 r^2 s}{q^2 r s}\)
4. \(\dfrac{x^2 y^2 z}{x^2 y^2 z}\)
5. \(\dfrac{m^3 n^2 o}{m^2 n^2 o}\)
### Detailed Solution Guide:
1. \(\dfrac{r^6}{r}\)
- Use the property of exponents \(\dfrac{a^m}{a^n} = a^{m-n}\).
- Simplify: \(r^{6-1} = r^5\).
2. \(\dfrac{s^3 t^4}{s^2 t^2}\)
- Apply the property of exponents to each variable separately.
- Simplify \(s\): \(s^{3-2} = s^1 = s\).
- Simplify \(t\): \(t^{4-2} = t^2\).
- Combined result: \(st^2\).
3. \(\dfrac{q^3 r^2 s}{q^2 r s}\)
- Apply the property of exponents to each variable separately.
- Simplify \(q\): \(q^{3-2} = q^1 = q\).
- Simplify \(r\): \(r^{2-1} = r\).
- Simplify \(s\): The \(s\) terms cancel out.
- Combined result: \(qr\).
4. \(\dfrac{x^2 y^2 z}{x^2 y^2 z}\)
- Any expression divided by itself equals 1.
- Simplified result: 1.
5. \(\dfrac{m^3 n^2 o}{m^2 n^2 o}\)
- Apply the property of exponents to each variable separately.
- Simplify \(m\): \(m^{3-2} = m\).
- Simplify \(n\): The \(n\) terms cancel out.
- Simplify \(o\
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