Fraction

Definition

A fraction is a numerical quantity that represents a part of a whole. It is expressed as a ratio of two integers, with the numerator representing the part and the denominator representing the whole.

5 Must Know Facts For Your Next Test

1. Fractions can be used to represent parts of a whole, ratios, and division.
2. Fractions can be added, subtracted, multiplied, and divided, following specific rules and procedures.
3. Simplifying fractions involves reducing the numerator and denominator to their lowest common factors.
4. Rational exponents, such as $\frac{1}{2}$ or $\frac{3}{4}$, can be used to represent fractional powers.
5. The laws of exponents, such as $x^{\frac{1}{2}} = \sqrt{x}$, apply to rational exponents.

Review Questions

• Explain how fractions can be used to represent parts of a whole, ratios, and division.
  • Fractions can be used to represent parts of a whole by expressing the relationship between the part and the whole. For example, $\frac{1}{4}$ represents one part out of four equal parts of a whole. Fractions can also be used to represent ratios, which are the relationship between two quantities, such as $\frac{3}{5}$ representing the ratio of 3 to 5. Additionally, fractions can be used to represent division, where the numerator is the dividend and the denominator is the divisor, such as $\frac{6}{3}$ representing the division of 6 by 3.
• Describe the process of simplifying fractions and how it relates to rational exponents.
  • Simplifying fractions involves reducing the numerator and denominator to their lowest common factors, which results in the most simplified form of the fraction. This process is similar to simplifying rational exponents, where the numerator and denominator of the exponent are reduced to their lowest common factors. For example, $x^{\frac{1}{2}}$ can be simplified to $\sqrt{x}$, and $x^{\frac{3}{4}}$ can be simplified to $\sqrt[4]{x^3}$. The laws of exponents, such as $x^{\frac{1}{n}} = \sqrt[n]{x}$, apply to rational exponents and help in simplifying fractional expressions.
• Analyze the relationship between fractions and rational exponents, and explain how they are interconnected in the context of simplifying expressions.
  • Fractions and rational exponents are closely related, as they both represent the same underlying mathematical concept. Rational exponents can be viewed as a generalization of fractions, where the numerator and denominator of the fraction are used as the exponent. This relationship allows for the application of the laws of exponents to simplify expressions involving fractions. For example, simplifying $x^{\frac{3}{4}}$ involves applying the rule $x^{\frac{a}{b}} = \sqrt[b]{x^a}$, which reduces the fractional exponent to a simpler form. The interconnection between fractions and rational exponents enables efficient simplification of complex expressions, making it a crucial concept in mathematics.

Related terms

Numerator: The numerator is the number on top of a fraction, which represents the part of the whole being considered.

Denominator: The denominator is the number on the bottom of a fraction, which represents the whole that is being divided into parts.

Proper Fraction: A proper fraction is a fraction where the numerator is less than the denominator, representing a part of the whole that is less than the whole.