Chapter 4 - Pre-Algebra 2: Fractions, Factors, and Exponents
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
Help!
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| Rules for divisibility | 2 - ends in an even number - 0, 2, 4, 6, or 8
3 - the sum of the digits can be divided by 3 evenly
4 - the last two digits can be evenly divided by 4
5 - ends in 0 or 5
6 - if divisible by 2 and 3
8 - last 3 digits can be divided by 8 evenl
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| Rules for divisibility (continued) | 9 - the sum of the digits can be divided by 9 evenly
10 - ends in 0
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| Factors | Any number that can be evenly divided into another number.
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| Order of operations | PEMDAS
1. Parentheses
2. Exponents
3. Multiply
4. Divide
5. Add
6. Subtract
Sentence: *Please Excuse My Dear Aunt Sally.*
*Multiply and divide, add and subtract left to right.*
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| Exponent | A number written to a power where the power (exponent) tell us how many times to multiply the number by itself.
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| Base | Example: 2^3
2 is the base.
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| Power | Example: 2^3
3 is the power, or exponent.
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| (-3)^2 means | (-3)^2 means
(-3)x(-3), which would equal 9.
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| -3^2 means | -3^2 means
3x3, which would equal -9 because you add the - sign to the answer once you have completed the problem, no matter what the number(s) is/are.
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| Prime Numbers | Numbers with only 2 factors; 1 and itself.
Example: 5=5x1
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| Composite Numbers | Numbers with 3 or more factors.
Example: 18=1, 2, 3, 6, and 9
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| GCF | Greatest Common Factor
The greatest number that is divisible into a group of numbers.
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| Variable GCF | Example: x^2y^5,xy^3
The answer is xy^3.
RULE: *Smallest power of common factors.*
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| Prime Factorization | Example: 120
2x60
2X30
2x15
5x3
The prime factorization would be 2^3x3x5.
*Hint: Use prime numbers in tree to determine prime factorization.*
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| Finding prime factorization using GCF | Example: 78 and 124
78
2x39
3x13
The prime factorization for 78 would be 2x3x13.
124
2x62
2x31
The prime factorization for 124 would be 2^2x31.
You then need to find what the two prime factorizations have in common with each oth
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| Finding prime factorization using GCF (continued) | FINAL ANSWER: The final answer is 2, because you must use the smallest power of common factors, which is 2, not 2^2.
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| What is the rule for solving problems involving the Greatest Common Factor of 2 or more numbers? | Always use the smallest power of common factors.
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| Simplifying Variable expressions | Example: 12x^2y^5
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8x^4y^2
The answer would be:
3y^3
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2x^2
You are simplifying what is already there.
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| What is the rule for simplifying variable expressions? | When you are simplifying variables with exponents, you subtract the smaller power from the larger power, and whatever that power is, that will be the power for your variables in your final answer.
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| Rational Numbers | Any number that can be written in the form a
-
b
where b ≠ (is not equal to) 0.
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| Examples of rational numbers | -2
-
8, 8
-
8, and -0.5.
Pi, or 3.14, (π) is not a rational number.
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| Negative Rational Numbers - how do you write | If the numerator is a positive number and the denominator is negative number, you make the numerator a negative number and the denominator a positive number; then, you simplify or reduce to the simplest (most simplified) term.
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| Negative Rational Numbers - how do you write (continued) | If both the numerator and the denominator are negative numbers, you make the entire fraction or term positive.
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| Multiplying exponent rule | Rule: Multiplying powers with the same base: a^m x a^n = a^m+n
and a^4 x a^7 = a^4+7.
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| Multiplying exponent rule Example | Example: 4x^3y^5 x 3x^2yz^2
12 x^3+2 y^5+1 z^2
Your final answer is: 12 x^5 y^6 z^2.
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| Power of Power rule | Rule: When multiplying a power of a power: (a^m)^n = a^mxn => (x^6)^4 = x^6x4 = x^24.
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| Power of Power rule Example | Example: (x^3y^5)^4
x^3x4 y^5x4
Your final answer is: x^12 y^20.
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| Zero exponent rule | Any number written to the zero power is 1.
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| Zero exponent rule Example | Example: x^0 = 1.
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| Negative exponent rule | Negative exponents are the reciprocal of the positive exponents.
Numerator = Denominator
Denominator = Numerator
(Swap negative numbers)
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| Negative exponent rule Example | Example: x^-3 = 1
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x^3.
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| Write without negative exponents (example) | Example: x^-3 y^2
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a^2 b^-3
Swap negatives!
The final answer is: y^2 b^3
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x^3 a^2.
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| Rewrite without a fraction bar | The number and variables with powers on the top (numerator{s}) remain the same. The numbers and variables with powers on the bottom (denominator{s}) are the opposite of themselves and are last in the fraction without a bar; (equation).
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| Rewrite without a fraction bar Example | Example: x^3 y^2
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a^2 b^3
Your final answer would be: x^3 y^2 a^-2 b^-3, as you kept the numerator(s) the same and made the denominator(s) the opposite (the numbers) of themselves.
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| Rewrite without zero or negative exponents and simplify | If there a variable with a zero exponent, you would never include it in your final answer. Anything with a zero exponent equals 1, and the number 1 is never necessary to be included in our final answer.
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| Rewrite without zero or negative exponents and simplify (continued) | This also applies for a variable by itself with no power/exponent by itself.
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| Rewrite without zero or negative exponents and simplify Example | Example: x^-3 y^2 z^0
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x^-2 y^-3
Rewrite: y^2 x^2 y^3
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x^3-2
Your final answer would be: y^5 over x.
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| Rewrite without zero or negative exponents and simplify Example (Continued) | BE SURE TO WRITE WITH A FRACTION (OVER)! You would not include z^0 or x^1; instead, you would include just x, as x^1 is equal to x.
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| Rewrite without zero or negative exponents and simplify | STEPS:
1) Rewrite without zero or negative exponents
2) Simplify
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| Austin O'Keefe | Period 2
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