Algebra: Properties of Equalities and Inequalities, Exponents, Radicals

Question 1

How do you understand closure property of real numbers?

Answer 1

Closure property:

• a + b is a unique real number
• a x b is a unique real number

Question 2

What is communitative property of real numbers?

Answer 2

Communitative property:

We can add or multiply numbers in any order.

• a* + b = b + a
• a* x b = b x a

Question 3

Describe associative property of real numbers.

Answer 3

Associative Property:

We can group numbers in different ways in a sum or in a product of real numbers. It doesn’t change the result.

• (a + b) + c = a +(b + c*)
• (ab)c = a(bc)*

Question 4

How do you understand identity property of real numbers?

Answer 4

Identity property:

Zero added to any real number equals the number itself.

a + 0 = a

Any real number multiplied by 1 is the number itself.

1(a) = a

Any real number multiplied by negative 1 is the opposite (or additive inverse) of itself.

-1(a) = -a

Question 5

How do you understand inverse property of real numbers?

Answer 5

Inverse property:

• a* + (-a) = 0
• a* x ¹/_a = 1

where a is not zero.

Question 6

Any real number multiplied by zero equals…..?

Answer 6

The product of any real number and zero is zero. This is property of zero.

a(0) = a

Question 7

• (-a) = ?
• (a + b) = ?
• (ab) = ?

Answer 7

• -(-a) = a*
• -(a + b) = -a + (-b)*
• -(ab) = (-a)b = a(-b)*

This is property of opposites for all real numbers.

Question 8

What does zero product property state for all real numbers?

Answer 8

The product of two real numbers a and b is zero (ab = 0) if and only if a = 0, b = 0, or both a and b = 0.

Question 9

How do you understand distributive property of real numbers?

Answer 9

It’s easy to understand and remember if you recall that “multiplication distributes over addition”.

a(b + c) = ab+ ac

Factoring out is a reverse operation to distribution over parentheses.

ab + ac = a(b + c)

Question 10

What is the definition of an axiom?

Answer 10

An axiom is a self-evident principle.

Question 11

What axioms of equality do you know?

Answer 11

These four following statements about equality are true for all real numbers a, b, and c.

• Reflexive property: Any number is equal to itself. a = a
• Symmetric property: If a = b, then b = a.
• Transitive property: If a = b and b = c, then a = c.
• Substitution property: if a = b, then a may replace b or b may replace a.

Question 12

What axioms of inequality do you know?

Answer 12

These are the axioms of inequality, also known as axioms of order.

• Trichotomy property:

For all real number a and b, one and only one of the following statements is true:

• a > b, a = b, or a*
• Transitive property:

For all real numbers a, b, and c:

If a >b and b > c, then a > c.

If a and b , then a .

Question 13

If the same number is added to equal numbers, the sums are…..?

Answer 13

The sums are equal.

Addition property : For all real numbers a, b, c, if the same number is added to equal numbers, the sums are equal.

If a = b, then a + c = b + c.

Question 14

If the same number is subtracted from equal numbers, the differences are…..?

Answer 14

The differences are equal.

Subtraction property: If the same number is subracted from equal numbers, the differences are equal.

If a = b, then a - c = b - c.

Question 15

If equal numbers are multiplied by the same nonzero number, the products are…..?

Answer 15

The products are equal.

This is multiplication property:

If a = b and c ≠ 0, then ac = bc.

If equal nuimbers are multiplied by the same nonzero number, the products are equal.

Question 16

If equal numbers are divided by the same nonzero number, the quotients are…..?

Answer 16

The quotients are equal.

This is division property:

If a = b and c ≠ 0, then ^a/_c = ^b/_c.

_{If equal numbers are divided by the same nonzero number, the quotients are equal.}

Answer 17

The following properties of inequalitites hold true for all real numbers, a, b, c, and d.

Addition

If a > b, then a + c > b + c.

If a < b, then a + c < b + c.

If a < b and c < d, then a + c < b + d.

If a > b and c > d, then a + c > b + d.

Answer 18

Subtraction

If a > b, then a - c > b - c.

If a < b, then a - c < b - c.

Answer 19

Multiplication

If a > b and c > 0, then ac > bc.

If a > b and c < 0, then ac < bc.

If 0 < a < b and 0 < c < d, then ac < bd.

Answer 20

Division

If a > b and c > 0, then a/_c > b/_c.

If a > b and c < 0, then a/_c < b/_c.

If a < b and ab > 0, then 1/_a > 1/_b

Answer 21

Some numbers may be written as the product of numbers that have identical factors.

Powers of a number can be written in factored form or in exponetial form:

• Factored form indicates the products of the factors as in b∙b∙b∙b∙b.
• Exponential form indicates the base and exponents as in b²

Question 22

What is scientific notation and where is it used?

Answer 22

Scientific notation is a way to express very large or very small numbers. You substitute zeros for power of 10 in exponential form.

Example:

6,000,000,000 = 6 x 10⁹ (count the number of places to the right of the first non-zero number)

0.00056 = 5.6 x 10^-4(count the number of places to the right of the decimal point, up to and including the first non-zero number)

Answer 23

Real numbers a and b. m and n are rational numbers.

For Multiplication: a^m ∙ aⁿ = a^m+n

For Division: a^m/_{a<span>ⁿ</span>} = am-n a ≠ 0

For a Power of a Product: (a^m)ⁿ + a^mn + (aⁿ)^m

For a Power of a Product: (ab)^m = a^m b^m

For a Power of a Quotient: (a/_b)^m = a^m/_b^m b ≠ 0

For a Zero Exponent: a⁰ = 1 a ≠ 0

For an Exponent of 1: a¹ = a

For a Negative Exponent: a^-n = 1/_aⁿ a ≠ 0

For a Base of 1: 1ⁿ = 1

Answer 24

Finding the square root of a number is the inverse of squaring a number. Since 5² = 25 and (-5)² = 25, the square root of 25 is both 5 and -5. Square root notation and properties follow.

• √ is called the radical sign.
• The number written beneath the radical sign is called the radicand. Example: √a.
• ^{<span><em>√a</em> is used to denote the principal or nonnegative square root of a positive real number <em>a</em>. </span>}
• -**√a is used to denote the negative square root of a positive real number a.
• ^{<span>±</span>} √a is used to denote the positive or negative square root of a positive real number a.
• The index of a root is the small number written above and to the left of the radical sign. it represents which root is to be taken. The index for square roots is 2. it is understood and therefore not included.
• (√a)² = a ^{where<em> a</em> is a real number.}
• √a² = **|a| where a is a real number.
• √a^2m = |a|^m whrere a is any real number, and m is any positive integer.
• 0 has only one square root, √0 = 0
• Negative numbers do not have square roots in the set of real numbers. See List 170, “Imaginary Numbers and Their Powers.
• In accordance wih the Product Property of Square Roots, √ab = √a √b, where a and b are nonnegative real numbers.
• In accordance with the Quotient Property of Square Roots, √a/_b = √a/_√b, where a is any nonnegative real number and b is a positive real number.
• In accordance with the Property of Square Roots of Equal Numbers, a² = b² if and only if a = b or a = -b, where a and b are real numbers.
• A radical is in simplest form when: no radicand has a square root factor (other than 1), the radicand is not a fraction, no radicals are in the denominator.
• Only radicals with like radicands may be added or subtracted.

Answer 25

• (ⁿ√a)ⁿ = a
• ⁿ√aⁿ = IaI if n is even or = a if n is odd
• ⁿ√ab = ⁿ√a ⁿ√b
• ⁿ√a/_b = ⁿ√a/_{<span><span><span>ⁿ</span></span></span>√b} b ¹ 0
• ^m√ _ⁿ√a = ^mn√a = ⁿ√_^m√a