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Magazine Chapter 1: Problem 73
Express as a quotient. $$x^{-3}+x^{2}$$
Short Answer
Expert verified
\(\frac{1 + x^5}{x^3}\)
Step by step solution
01
Understanding Exponents
Know that negative exponents means division. If you see an expression like \(x^{-n}\), it can be expressed as \(\frac{1}{x^n}\). Therefore, \(x^{-3}\) can be rewritten as \(\frac{1}{x^3}\).
02
Rewrite Negative Exponent as Fraction
Rewrite the term \(x^{-3}\) as a fraction: \(\frac{1}{x^3}\). This gives us \(\frac{1}{x^3} + x^2\).
03
Create a Common Denominator
In order to express the sum as a single fraction (quotient), we need to have a common denominator. The common denominator for \(\frac{1}{x^3}\) and \(x^2\) is \(x^3\).
04
Convert Whole Number to Fraction
Convert \(x^2\) to a fraction with the common denominator: \(x^2 = \frac{x^2 \cdot x^3}{x^3} = \frac{x^5}{x^3}\).
05
Combine Fractions
Now combine the two fractions: \(\frac{1}{x^3} + \frac{x^5}{x^3} = \frac{1 + x^5}{x^3}\).
06
Final Expression
The expression \(x^{-3} + x^2\) has been expressed as the single quotient \(\frac{1 + x^5}{x^3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fractions
Fractions are a way of representing parts of a whole. In mathematics, a fraction is composed of a numerator鈥攖he top part鈥攁nd a denominator鈥攖he bottom part. The numerator shows how many parts we have, while the denominator shows into how many equal parts the whole is divided.
For example, in the fraction \( \frac{3}{4} \), 3 is the numerator, and 4 is the denominator, which means we have 3 out of 4 total parts. Fractions are especially useful when you perform operations like addition, subtraction, multiplication, and division on numbers that are not whole.
With algebraic fractions, just like with numerical fractions, you often need to find a common denominator to perform arithmetic operations. This helps in simplifying calculations and combining expressions.
For example, in the fraction \( \frac{3}{4} \), 3 is the numerator, and 4 is the denominator, which means we have 3 out of 4 total parts. Fractions are especially useful when you perform operations like addition, subtraction, multiplication, and division on numbers that are not whole.
With algebraic fractions, just like with numerical fractions, you often need to find a common denominator to perform arithmetic operations. This helps in simplifying calculations and combining expressions.
Common Denominator
When adding or subtracting fractions, having a common denominator is necessary to combine the fractions into a single expression. It simply means finding a shared denominator for all fractions involved in the operation. The denominators should be the same to perform these operations smoothly.
Take for example, the fractions \( \frac{1}{3} \) and \( \frac{1}{4} \). In this case, the least common denominator would be 12. You would adjust each fraction to have this common denominator: \( \frac{1}{3} = \frac{4}{12} \) and \( \frac{1}{4} = \frac{3}{12} \).
By having a common denominator, fractions can be easily added or subtracted:
Take for example, the fractions \( \frac{1}{3} \) and \( \frac{1}{4} \). In this case, the least common denominator would be 12. You would adjust each fraction to have this common denominator: \( \frac{1}{3} = \frac{4}{12} \) and \( \frac{1}{4} = \frac{3}{12} \).
By having a common denominator, fractions can be easily added or subtracted:
- \( \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \)
- \( \frac{4}{12} - \frac{3}{12} = \frac{1}{12} \)
Simplifying Expressions
Simplifying an expression involves rewriting it in its simplest form. For fractions, this usually means reducing them so that the numerator and the denominator have no common factors other than 1.
In algebra, simplifying expressions often involves combining like terms or factoring. For example, if you have \( \frac{2x + 4}{x + 2} \), you could factor the numerator: \( 2(x + 2) \), and cancel out the common factors across the fraction, resulting in the simplified expression \( 2 \).
Simplifying helps make expressions easier to understand and work with, especially when you need to solve equations or evaluate expressions. It's particularly useful in a scenario where you have a complex fraction, and you need to reduce it to discover the most straightforward form.
Always remember: making an expression simpler is not changing its value, but just its appearance, which makes mathematical operations easier to handle.
In algebra, simplifying expressions often involves combining like terms or factoring. For example, if you have \( \frac{2x + 4}{x + 2} \), you could factor the numerator: \( 2(x + 2) \), and cancel out the common factors across the fraction, resulting in the simplified expression \( 2 \).
Simplifying helps make expressions easier to understand and work with, especially when you need to solve equations or evaluate expressions. It's particularly useful in a scenario where you have a complex fraction, and you need to reduce it to discover the most straightforward form.
Always remember: making an expression simpler is not changing its value, but just its appearance, which makes mathematical operations easier to handle.
