Algebra Honors ( Periods 4 & 7)

Multiplication Properties of Exponents 7-1

Monomial: A number, variable, or product of a number and variables with NON-NEGATIVE INTEGER EXPONENTS.

Negative integer exponents mean that a variable is in the denominator. Numbers can be in the denominator.

Integer exponents mean that you can’t have variables with square roots or other roots (these would be fractional exponents)

Therefore, monomials are ONE TERM and terms are separated by addition or subtraction NOT multiplication or division.

A constant is a monomial that is a real number (it can be negative, positive, a fraction/decimal or in a radical sign)

Linear expressions have each variable to the 1 power. Nonlinear expressions have either one variable to a power of 2 or more OR multiple variables attached together.

The base of an exponent is the number or variable at the bottom.

The exponent is the little superscript number at the top.

This is called exponential form or a power of the base raised to the specific exponent.

Expanded form is when you show the repeated multiplication.

Standard or simplified form is the number answer (simplified for a variable is the same as the exponential)

3⁴ is exponential form or we say it’s 3 raised to the power of 4

3●3●3●3 is expanded form

81 is standard form

ODD/EVEN RULES WITH POWERS AND NEGATIVE:

An odd number of negatives = negative

An even number of negatives = positive

So

An odd power with a negative integer in a ( ) = negative

An even power with a negative integer in a ( ) = positive

BUT

An EVEN power with a negative power WITHOUT ( ) = NEGATIVE…there’s only ONE negative here and that’s odd…I call this negative the ZAPPER because it zaps the answer at the very end of the simplifying

An odd power without ( ) = still negative…there’s only ONE negative here and that’s odd

EXAMPLES:

EVALUATING WITH POWERS:

If you are given a variable to a power, you simply plug and chug the value given for the variable.

USUALLY YOU SHOULD PLACE THE VARIABLE IN ( ) WHEN PLUGGING IN:

a² – b⁴ if a = 2 and b =3

(2) ² – (3) ⁴ = 4 - 81 = -77

VS this case where you’re doing an operation INSIDE the (  ) FIRST, then doing the power:

(a – b) ⁴ = (2 – 3) ⁴ = (-1)⁴ = 1

Algebra ( Period 1)

Multiplication Properties of Exponents 7-1

Monomial: A number, variable, or product of a number and variables with NON-NEGATIVE INTEGER EXPONENTS.

Negative integer exponents mean that a variable is in the denominator. Numbers can be in the denominator.

Integer exponents mean that you can’t have variables with square roots or other roots (these would be fractional exponents)

Therefore, monomials are ONE TERM and terms are separated by addition or subtraction NOT multiplication or division.

A constant is a monomial that is a real number (it can be negative, positive, a fraction/decimal or in a radical sign)

Linear expressions have each variable to the 1 power. Nonlinear expressions have either one variable to a power of 2 or more OR multiple variables attached together.

The base of an exponent is the number or variable at the bottom.

The exponent is the little superscript number at the top.

This is called exponential form or a power of the base raised to the specific exponent.

Expanded form is when you show the repeated multiplication.

Standard or simplified form is the number answer (simplified for a variable is the same as the exponential)

3⁴ is exponential form or we say it’s 3 raised to the power of 4

3●3●3●3 is expanded form

81 is standard form

ODD/EVEN RULES WITH POWERS AND NEGATIVE:

An odd number of negatives = negative

An even number of negatives = positive

So

An odd power with a negative integer in a ( ) = negative

An even power with a negative integer in a ( ) = positive

BUT

An EVEN power with a negative power WITHOUT ( ) = NEGATIVE…there’s only ONE negative here and that’s odd…I call this negative the ZAPPER because it zaps the answer at the very end of the simplifying

An odd power without ( ) = still negative…there’s only ONE negative here and that’s odd

EXAMPLES:

EVALUATING WITH POWERS:

If you are given a variable to a power, you simply plug and chug the value given for the variable.

USUALLY YOU SHOULD PLACE THE VARIABLE IN ( ) WHEN PLUGGING IN:

a² – b⁴ if a = 2 and b =3

(2) ² – (3) ⁴ = 4 - 81 = -77

VS this case where you’re doing an operation INSIDE the (  ) FIRST, then doing the power:

(a – b) ⁴ = (2 – 3) ⁴ = (-1)⁴ = 1

Math 6H ( Period 5)

Equivalent Ratios in the Real World 1-6

There are 2 ways to determine if 2 ratios are equivalent:

1. Restate them both as unit rates. If they’re the same unit rates, they’re equivalent.

2. Use equivalent fractions if finding the unit rate is difficult (let’s say you need to go several decimal places in the division and don’t have a calculator)

Example using both methods:

You know that one store will charge you $2 for 10 photo prints and another has a sign offering 30 prints for $6.

Are these 2 stores charging EQUIVALENT AMOUNTS?

First way: Unit Rates

The unit rates are the same:

You can either find the prints per dollar or the unit price per print.

Both will show that the stores are equivalent.

Prints per dollar:

10 prints/$2 = 5 prints/$1 and 30 prints/$6 = 5 prints/$1 as well

Unit price:

$2/10 prints = $.20/print and $6/30 prints = $.20/print as well

Second way: Equivalent Fractions

The equivalent fraction approach asks you to see if you can find a common factor that the numerator and denominator were both either multiplied by or divided by to get to each other:

Can I get from 10/2 to 30/6 by multiplied both the numerator and denominator by the same number?

Yes I can!

I can multiply by 3 (scale forward by 3)

When I do this, it’s helpful to show a curved arrow with the x 3 on each arrow:

So by using equivalent fractions, the 2 stores are offering the same deal.