Algebraic Notations and Symbolic Structures

Meaning of Induction

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Mathematical induction is commonly used to prove statements involving:

Fractions

Natural Numbers

Triangles

Decimals

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Algebraic Notation

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Select all valid algebraic notations.

x+y

3=3

\(a^2\)

circle

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Explanation

Expression or Identity

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\(a+b=b+a\) is an example of an algebraic (1).

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identity

fraction

prime

equation error

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Explanation

Simplify Expression

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Simplify: \(2x + 3x\)

5x

6x

x

\(x^2\)

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Recognize an Identity

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Which statement is true for all values of x?

x+1=5

2x=8

x+x=2x

x-3=0

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Order of Terms

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Please drag and drop the options to sort them

Arrange the algebraic structures from simplest to most complex.

Identity

Variable

Expression

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Explanation

Matching Expressions

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Match the algebraic expression with its meaning.

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Click on an item in the first group
Click on the match in the second group

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Algebric Expression

x+x+x

\(a\times a\)

Meaning

\(a^2\)

3x

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Classify Algebraic Forms

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Place each item in the correct category.

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Power Notation

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x3 means x multiplied by itself (1) times.

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2

3

4

5

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Equivalent Expressions

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Which expression is equivalent to \(4(a+2)\)?

4a+2

4a+8

a+8

8a

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Explanation

Factorial Notation

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What is the value of \(7!\)?

49

720

5040

40320

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Explanation

Zero Factorial

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Based on the definition derived from permutation notation, what is the value of \(0!\)?

0

1

Undefined

\(\infty\)

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Explanation

Permutation Formula

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The number of permutations (arrangements) of \(n\) different things taken \(r\) at a time is given by \({}^nP_r =\) .

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Combination Formula Value

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Calculate \({}^7C_3\).

21

35

42

210

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Algebraic Identity: Sum of Squares

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Which of the following is a correct algebraic identity?

\((a+b)^2 = a^2 + b^2\)

\((a+b)^2 = a^2 + 2ab + b^2\)

\((a+b)^2 = a^2 - 2ab + b^2\)

\((a+b)^2 = a^2 + b^2 - ab\)

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Explanation

Sum of First n Natural Numbers

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Gauss showed that the sum \(S = 1 + 2 + 3 + \cdots + n\) can be expressed as a closed formula. Enter this formula below.

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Explanation

Match Algebraic Identities

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Match each algebraic expression on the left with its equivalent identity on the right.

To complete the line match

Click on an item in the first group
Click on the match in the second group

To delete a match, double click on a line

Expression

\(a^3 + b^3\)

\(a^3 - b^3\)

\((a+b)^2 - (a-b)^2\)

\(a^2 - b^2\)

Identity

\((a+b)(a-b)\)

\(4ab\)

\((a+b)(a^2 - ab + b^2)\)

\((a-b)(a^2 + ab + b^2)\)

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Explanation

Binomial Expansion: Number of Terms

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When expanding \((x + y)^n\) using the Binomial Theorem (where \(n\) is a natural number), how many terms does the expansion contain?

\(n\) terms

\(n - 1\) terms

\(n + 1\) terms

\(2n\) terms

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Value of \(a^0\)

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For any non-zero value of \(a\), what is the value of \(a^0\)?

0

1

\(a\)

Undefined

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Explanation

Sum of Cubes Formula

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Enter the closed-form formula for \(\displaystyle\sum_{k=1}^{n} k^3 = 1^3 + 2^3 + 3^3 + \cdots + n^3\).

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Meaning of Induction

Correct Answer

Explanation

Algebraic Notation

Correct Answer

Explanation

Expression or Identity

Correct Answer

Explanation

Simplify Expression

Correct Answer

Explanation

Recognize an Identity

Correct Answer

Explanation

Order of Terms

Correct Answer

Explanation

Matching Expressions

Algebric Expression

Meaning

Correct Answer

Explanation

Classify Algebraic Forms

drag and drop the selected option to the right place

Correct Answer

Explanation

Power Notation

Correct Answer

Explanation

Equivalent Expressions

Correct Answer

Explanation

Factorial Notation

Correct Answer

Explanation

Zero Factorial

Correct Answer

Explanation

Permutation Formula

Correct Answer

Explanation

Combination Formula Value

Correct Answer

Explanation

Algebraic Identity: Sum of Squares

Correct Answer

Explanation

Sum of First n Natural Numbers

Correct Answer

Explanation

Match Algebraic Identities

Expression

Identity

Correct Answer

Explanation

Binomial Expansion: Number of Terms

Correct Answer

Explanation

Value of \(a^0\)

Correct Answer

Explanation

Sum of Cubes Formula

Correct Answer

Explanation

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