Reasoning Problems Flashcards
How do you understand identity property of real numbers?
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
How do you understand inverse property of real numbers?
Inverse property:
- a* + (-a) = 0
- a* x 1/a = 1
where a is not zero.
Any real number multiplied by zero equals…..?
The product of any real number and zero is zero. This is property of zero.
a(0) = 0
- (-a) = ?
- (a + b) = ?
- (ab) = ?
- -(-a) = a*
- -(a + b) = -a + (-b)*
- -(ab) = (-a)b = a(-b)*
This is property of opposites for all real numbers.
What does zero product property state for all real numbers?
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.
How do you understand distributive property of real numbers?
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)
Real numbers a and b. m and n are rational numbers.
For Multiplication: am ∙ an **= **am+n
For Division: am/a<span>n</span> = am-n a ≠ 0
For a Power of a Product: (am)n + amn + (an)m
For a Power of a Product: (ab)m = am bm
For a Power of a Quotient: (a/b)m = am/bm 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
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.
- √a2m = |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.
- (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
When you multiply more than two signed numbers, when is their product positive?
When you multiply signed numbers, find the product of the absolute values of all factors.
The product is positive when:
- all factors are positive
- there is an even number of negative factors.
Example:
(-5) x (2) x (-1) x (2) = 20
When you multiply more than two signed numbers, when is the product negative?
Find the product of the absolute values of all factors.
The product is negative when there is an odd number of negative factors.
Example:
(-5) x (-2) x (-1) x (2) = -20
