Illustrative Math Alignment: Grade 6 Unit 9

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to the topics of rationals and radicals. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to to the topic of fractions. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to to the topic of fractions. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to to the topic of fractions. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to to the topic of fractions. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to to the topic of polynomials. Note: The download is an MP4 video.

This is part of a collection of math video definitions related to to the topic of polynomials. Note: The download is an MP4 video.

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This is part of a collection of definitions on the topic of Vectors.

In this tutorial students learn about rational expressions and use the properties of fractions to combine them.

—Press PREVIEW to see the tutorial—

Topic

Calculus

Definition

Acceleration is the rate of change of velocity with respect to time. It is the second derivative of position with respect to time.

Description

Acceleration is a fundamental concept in calculus, particularly in the study of motion and kinematics. It's crucial for understanding how objects move and change speed over time. In real-world applications, acceleration is used extensively in physics and engineering, from designing roller coasters to launching spacecraft. For example, in automotive engineering, understanding acceleration helps in designing safer vehicles and more efficient engines.

Topic

Calculus

Definition

The derivative of a rational function f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0, is given by the quotient rule: f'(x) = (Q(x)P'(x) - P(x)Q'(x)) / [Q(x)]2.

Description

The derivative of rational functions is a fundamental concept in calculus with significant applications in various fields of science and engineering. It's particularly useful in analyzing rates of change in complex systems, solving differential equations, and optimizing processes in fields like economics, physics, and engineering.

Topic

Calculus

Definition

Integration by substitution is a method used to evaluate integrals by substituting a new variable for a part of the integrand, often simplifying the integral.

Description

Integration by substitution is a fundamental technique in calculus, often considered the counterpart to the chain rule for derivatives. It's particularly useful for integrating composite functions and functions involving trigonometric, exponential, or logarithmic terms. This method is widely applied in physics and engineering to solve problems involving changing variables or coordinate systems.

Topic

Calculus

Definition

A rational function is a function that can be expressed as the ratio of two polynomials, P(x)/Q(x), where Q(x) ≠ 0.

Description

Rational functions are a crucial class of functions in calculus and algebra, with wide-ranging applications in science, engineering, and economics. They exhibit interesting behaviors such as asymptotes, holes, and discontinuities, making them valuable for modeling complex real-world phenomena. The study of rational functions involves analyzing their domain, range, intercepts, and end behavior.

Topic

Rationals and Radicals

Definition

The fourth root of a number is a value that, when raised to the power of four, gives the original number.

Description

The fourth root is an important concept in Radical Numbers, Expressions, Equations, and Functions. It is the inverse operation of raising a number to the power of four. Understanding fourth roots is crucial for solving higher-degree equations and for simplifying radical expressions.

Fourth roots are also relevant in various scientific and engineering applications where higher-degree roots are required. They are a part of advanced mathematical studies, including algebra and calculus.

Topic

Rationals and Radicals

Definition

The geometric mean of two numbers is the square root of their product.

Description

The geometric mean is an example of using radical expressions to solve a problem. It provides a measure of central tendency that is particularly useful in situations where the numbers are multiplied together rather than added.

In geometry, the geometric mean appears in various theorems and constructions. It is also used in finance and statistics to calculate average growth rates and to compare different sets of data.

Topic

Rationals and Radicals

Definition

Rationalizing the denominator is the process of eliminating radicals or complex numbers from the denominator of a fraction by multiplying both the numerator and denominator by an appropriate factor.

Description

Rationalizing the Denominator is a crucial technique in the study of Radical Numbers, Expressions, Equations, and Functions. This process involves removing radicals or complex numbers from the denominator of a fraction, which simplifies the expression and often makes it easier to evaluate or compare with other expressions. For example, to rationalize the denominator of 

Topic

Rationals and Radicals

Definition

The conjugate of a radical expression is obtained by changing the sign between two terms in a binomial.

Description

In the context of Radical Numbers, Expressions, Equations, and Functions, the concept of conjugates is essential for simplifying expressions. When dealing with radicals, particularly in the denominator, multiplying by the conjugate can help to rationalize the expression. This process eliminates the radical from the denominator, making the expression easier to work with.

Topic

Rationals and Radicals

Definition

Rationalizing a radical is the process of eliminating radicals from the denominator of a fraction by multiplying both the numerator and denominator by an appropriate factor.

Description

Rationalizing a Radical is an important technique in the study of Radical Numbers, Expressions, Equations, and Functions. This process involves eliminating radicals from the denominator of a fraction, which simplifies the expression and often makes it easier to work with or compare to other expressions. For example, to rationalize the denominator of 

$$\frac{1}{\sqrt{3}}$$