MrE's Algebra Blog

Algebra: Chapter 11, Lesson 6, page 504.

Addition and Subtraction of Radical Expressions

To add and subtract radical expressions, you can use the same distributive property we have used in the past. You may have to rationalize the denominator and after that, you might be able to factor and combine expressions.

Follow along:

`sqrt(3) + sqrt((1/3)) = sqrt(3) + sqrt(1)/sqrt(3)`

now we have to rationalize the second term, by multiplying by `1/1` or `sqrt(3)/sqrt(3)`

= `sqrt(3) + sqrt(1)/sqrt(3) * sqrt(3)/sqrt(3)`, and we remember (see the denominator) that `sqrt(3)*sqrt(3) = 3`

= `1sqrt(3) + 1[sqrt(3)]/3` remember too, that 1+ 1/3 = 4/3,

then finally we have our solution = `4/3 sqrt(3)`

The best way is to see some more examples, click here for them!

Here again is a great link from Purplemath.

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 9, Lesson 3, page 411.

Equations and Absolute Value

To solve an equation of the form `| A | = b`, solve the disjunction `A = b` OR `A = −b`. You will have 2 equations to solve with the right side of the second equation having the opposite sign of the first equation’s right side.

REMEMBER by definition, the solution of `| A | ≠ a` NEGATIVE NUMBER! So … the solution to these type of problems is the NULL SET! or the symbol `∅` !

Here is a link to examples!

Two of tonight’s homework problems solved by MrE are here! Just click it!

SNOW DAY #4!

Algebra: Chapter 11, Lesson 5, page 498.

Dividing and Simplifying

The `sqrt` of quotients is pretty simple. You can combine or break apart quotient `sqrt`s to your liking.

Try to find perfect squares and make sure that all the factors are simplified.

Division Property for Radicals:

`sqrt(a/b) = sqrt(a)/sqrt(b)` Remember too, that you can go back and forth without any problem.

Remember too, to “rationalize the denominator”. Make sure that NO radical appears in the denominator. If you have one, multiply the numerator and denominator by 1 (the `sqrt` of the denominator) to make it disappear. The `sqrt(a) * sqrt(a) = sqrt(a^2) = a`!

An expression containing radicals is simplified when the following conditions are met:

• The radicand contains NO perfect square factors
• A fraction in simplest form does not have a radical in the denominator
• A simplified radical does not contain a fractional radicand.

For example:

`sqrt(2)/sqrt(3) = sqrt(2)/sqrt(3) * sqrt(3)/sqrt(3)`

`= [sqrt(2) * sqrt(3)] / [sqrt(3) * sqrt(3)]`

`= sqrt(6)/3 = [1/3] * sqrt(6)`

See this link from purplemath.com.

Here again is a great link from Purplemath.

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 9, Lesson 2, page 405.

Compound Sentences

Conjunctions and Disjunctions are like Intersections and Unions (respectively) from lesson 9-1.

A disjunction of 2 statements is formed by connecting them with the word “or”. A disjunction is true when one OR both statements are true.

A conjunction of 2 statements is formed by connecting them with the word “and”. A conjunction is true when both statements are true.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra: Chapter 11, Lesson 4, page 495.

Multiplying Radical Expressions

Remember that the `sqrt(ab) = sqrt(a)⋅sqrt(b)` and you can’t go wrong. Make sure that you simplify and identify perfect squares. Practice makes perfect. The steps can be stated as:

• Multiplying
• Factoring to find perfect square factors
• Identifying perfect squares
• Simplify

For example:

`sqrt(3x^2)⋅sqrt(9x^3)`

This becomes `sqrt(3⋅9⋅x^5) = sqrt(3⋅9⋅x^4⋅x)`

re-arraigning terms it looks like =`sqrt(9)⋅sqrt(x^4)⋅sqrt(3)⋅sqrt(x)`

and that finally simplifies to = `3x^2⋅sqrt(3x)`.

Click this purplemath.com link for some more explanation and practice!

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 9, Lesson 1, page 400.

Sets, Intersections and Unions

A set is a well-defined collection of objects called members or elements.

• Roster notation LISTS the members of the set.
• Set-Builder Notation gives a DESCRIPTION of how the set is built.

The intersection of 2 sets `A` and `B`, written `A ∩ B` is the set of all members that are COMMON to both sets. We say ” A intersection B”.

The union of 2 sets `A` and `B`, written `A ∪ B` is the set of all members that are in `A` or `B` or in both. If an intersection is EMPTY, we say the intersection is the empty set which is symbolized as `∅`.

All of these concepts are described here too with examples!

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra: Chapter 11, Lesson 3, page 491.

Simplifying Radical Expressions

`sqrt(x^2)` = square root `(x^2)` = `x`

Its easy to simplify radicals. You can break up numbers and variables, because multiplication is commutative. If asked to find the `sqrt(100)` , we could break up `100` into `25 * 4`. We know that the `sqrt(25) = 5` and the `sqrt(4) = 2`, then the `sqrt(100) = 5 * 2 = 10`. You can do the same with variables that have exponents.

If asked to find the `sqrt` of a variable with even exponents, `sqrt(x^6)` for example, the answer is just the variable with the exponent divided in 2. So for `sqrt(x^6)`, the answer is `x^3`. If the variable has odd exponents, like `x^27`, convert that to `(x^26)*(x^1)` and then take the `sqrt(x^26) * sqrt(x^1) = (x^13)* sqrt(x)`.

See these examples (1/2 way down the page) too.

Here again is a great link from Purplemath.

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 9, Lesson 1, page 400.

Sets, Intersections and Unions

A set is a well-defined collection of objects called members or elements.

• Roster notation LISTS the members of the set.
• Set-Builder Notation gives a DESCRIPTION of how the set is built.

The intersection of 2 sets `A` and `B`, written `A ∩ B` is the set of all members that are COMMON to both sets. We say ” A intersection B”.

The union of 2 sets `A` and `B`, written `A ∪ B` is the set of all members that are in `A` or `B` or in both. If an intersection is EMPTY, we say the intersection is the empty set which is symbolized as `∅`.

All of these concepts are described here too with examples!

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra: Chapter 11, Lesson 1 and Lesson 2, page 482 and page 487.

Real Numbers (Square Roots) and Radical Expressions

Definition: the number `c` is a square root of `a` if `c^2=a`. In math symbols then, `c = sqrt(a^2)`

Prinicipal square root is the positive square root of a number, like `sqrt(36)=6`

Real numbers have 2 sets, the rational numbers and the irrational numbers.

• Rational number can be expressed as a ratio of 2 integers. They can have a repeating decimal AS LONG AS THERE IS A PATTERN.
• Irrational numbers conversely, cannot be expressed as a ratio or have a repeating decimal WITH NO PATTERN. The best irrational number example is π.

We, in Algebra 1, cannot take the square root of a negative numbers. By definition, then all RADICANDS, the thing under the square root symbol, MUST always be positive.

An expression written under the radical is also called a radical expression. With the exception of perfect square numbers (0, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100 …) most square roots of whole numbers are irrational.

Finally, the `sqrt(a^2)` can be simplified to `| a |`, this gives up 2 values for the square root, a positive and negative value.

Here again is a great link from Purplemath.

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 9, Lesson 1, page 400.

Sets, Intersections and Unions

A set is a well-defined collection of objects called members or elements.

• Roster notation LISTS the members of the set.
• Set-Builder Notation gives a DESCRIPTION of how the set is built.

The intersection of 2 sets `A` and `B`, written `A ∩ B` is the set of all members that are COMMON to both sets. We say ” A intersection B”.

The union of 2 sets `A` and `B`, written `A ∪ B` is the set of all members that are in `A` or `B` or in both. If an intersection is EMPTY, we say the intersection is the empty set which is symbolized as `∅`.

All of these concepts are described here too with examples!

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra: Chapter 10 Benchmark!

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Algebra 1a : Chapter 8 Benchmark!

Algebra: Chapter 10 Review

Algebra Benchmark #10 Review (200 points) worked on in class and for homework!

Make sure that your notes are organized are ready to go!

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Algebra 1a : Chapter 8 Review

Test on Monday, make sure that your notes are all up-to-date. The Chapter 8 Review Packet for homework is a great prep for the test.

Make sure that you REMEMBER HOW TO GRAPH 2 EQUATIONS OR SUBSTITUTE ONE FOR ANOTHER OR ADD/SUBTRACT 2 EQUATIONS TO SOLVE FOR EITHER X OR Y.

Chapter 10, Lesson 12, page 475.

Reasoning Strategies

You can use the following strategies on the reasoning strategy problems in Chapter 10-12.

• Draw a diagram
• Make an organized list
• Look for a pattern
• Try, test and revise
• Use logical reasoning
• Simplify the problem
• Write an equation
• Make a table
• Work backward

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Algebra 1a : Chapter 8 Review

Test on Monday, make sure that your notes are all up-to-date. The Chapter 8 Review Packet for homework is a great prep for the test.

Make sure that you REMEMBER HOW TO GRAPH 2 EQUATIONS OR SUBSTITUTE ONE FOR ANOTHER OR ADD/SUBTRACT 2 EQUATIONS TO SOLVE FOR EITHER X OR Y.

Algebra: Chapter 10, Lesson 10, page 469 (day #2)

Complex Rational Expressions

To simplify a complex rational expression, multiply the numerator and denominator by an expression equivalent to `1`. The expression selected should use the least common multiple of any denominator found in the numerator or denominator of the complex rational expression.

Sometimes it is easier to just work with the numerator and the denominator separately AND THEN, combine them with their division. Problems like 23-29 are HARD, look at my solutions for my method. You may have a different approach and that is OK!

Here is a link from purplemath with more examples.

Two of tonight’s homework problems solved by MrE are here! Just click it

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Algebra: Chapter 8, Lesson 6, page 387 (day #2).

Digit and Coin word problems.

Just remember to write the coin problems with the d (dime), q (quarter), n (nickel) preceeded by the value of the coin remembering that the d, q or n stand for the number of that type of coin. For example, `.05n + .10d = 2.05`. You can then multiply both sides by `100` to clear the decimals.

Remember too, that any 2-digit number can be expressed as `10x + y` where `x` is the digit in the tens place and `y` is the digit in the one (units) place. For example, the number `23` can be written as `10 * 2 + 3`. If we reverse the digits in the original number, the new number can be expressed as `10y + x`. The reverse of `23`, `32` can be written as `10 * 3 + 2`.

Here is a link for some examples of coin problems and here is a link for digit type problems (about 1/2 the way down the page)!

Two of tonight’s homework problems solved by MrE are here! Just click it!