zero and negative exponents common core algebra 1 homework

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Algebra 1 - Worksheet 13 - Zero and Negative Exponents

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Negative Exponents and Zero Exponents

So far in this unit, you've learned how to simplify monomial expressions with positive exponents. Now we are going to study two more aspects of monomials: those that have negative exponents and those that have zero as an exponent .

I am going to let you investigate to see if you can come up with the rule on your own! Take a look at the following problems and see if you can determine the pattern.

Can you figure out the rule? If not, here it is...

The Rule for Negative Exponents:

The expression a -n is the reciprocal of a n

A reciprocal is when you "flip a fraction".

The reciprocal of 3/4 is 4/3.

The reciprocal of 5 is 1/5. (You can make a whole number a fraction by putting a one in the denominator: 5 = 5/1)

***An easy rule to remember is: if the number is in the numerator (top), move it to the denominator (bottom). If the number is in the denominator, move it to the numerator!

Let's take a look at a couple of examples:

Examples of Negative Exponents

Now let's quickly take a look at monomials that contain the exponent 0.

Any number (except 0) to the zero power is equal to 1.

Not too hard, is it? Let's look at a couple of example problems and then you can practice a few.

Example 1: Negative Exponents

Example 2: evaluating negative exponents.

**Since 2/3 is in parenthesis, we must apply the power of a quotient property and raise both the 2 and 3 to the negative 2 power.

First take the reciprocal to get rid of the negative exponent.

Then raise (3/2) to the second power.

Now, it's going to get a little more tough.

Example 3: Complex Expressions with Negative Exponents

One more example.

Example 4: More Negative Exponents

Yes, I know that's a lot of examples to comprehend. My goal was to start easy and progress to harder problems. Are you ready to try a few on your own?

Practice Problems

So, how did you do? Are you ready to move onto Scientific Notation ?

• Negative Exponents

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How to Teach Zero and Negative Exponents

Why is a 0  = 1?

Patterns with negative bases, patterns with negative exponents.

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Zero and Negative Exponents – PDFs

7-1 Assignment – Zero and Negative Exponents  (FREE) 7-1 Bell Work – Zero and Negative Exponents  (FREE) 7-1 Exit Quiz – Zero and Negative Exponents  (FREE) 7-1 Slide Show – Zero and Negative Exponents  (FREE) 7-1 Guide Notes SE – Zero and Negative Exponents  (FREE) 7-1 Guided Notes Teacher Edition – Zero and Negative Exponents ( Members Only ) 7-1 Lesson Plan – Zero and Negative Exponents ( Members Only ) 7-1 Online Activities – Zero and Negative Exponents ( Members Only ) 7-1 Video Lesson – Zero and Negative Exponents ( Members Only ) 7-1 Project – Zero and Negative Exponents ( Members Only )

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Study Guides > College Algebra CoRequisite Course

Zero and negative exponents, learning outcomes.

• Simplify expressions with exponents equal to zero.
• Simplify expressions with negative exponents.
• Simplify exponential expressions.

[latex]\dfrac{t^{8}}{t^{8}}=\dfrac{\cancel{t^{8}}}{\cancel{t^{8}}}=1[/latex]

A General Note: The Zero Exponent Rule of Exponents

Using order of operations with fractions.

[latex]\dfrac{5 a^m z^2}{a^mz}\quad=\quad 5\cdot\dfrac{a^m}{a^m}\cdot\dfrac{z^2}{z} \quad=\quad 5 \cdot a^{m-m}\cdot z^{2-1}\quad=\quad 5\cdot a^0 \cdot z^1 \quad=\quad 5z[/latex]

Example: Using the Zero Exponent Rule

• [latex]\dfrac{{c}^{3}}{{c}^{3}}[/latex]
• [latex]\dfrac{-3{x}^{5}}{{x}^{5}}[/latex]
• [latex]\dfrac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex]
• [latex]\dfrac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex]
• [latex]\begin{align}\frac{c^{3}}{c^{3}} & =c^{3-3} \\ & =c^{0} \\ & =1\end{align}[/latex]
• [latex]\begin{align} \frac{-3{x}^{5}}{{x}^{5}}& = -3\cdot \frac{{x}^{5}}{{x}^{5}} \\ & = -3\cdot {x}^{5 - 5} \\ & = -3\cdot {x}^{0} \\ & = -3\cdot 1 \\ & = -3 \end{align}[/latex]
• [latex]\begin{align} \frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}& = \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{1+3}} && \text{Use the product rule in the denominator}. \\ & = \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}} && \text{Simplify}. \\ & = {\left({j}^{2}k\right)}^{4 - 4} && \text{Use the quotient rule}. \\ & = {\left({j}^{2}k\right)}^{0} && \text{Simplify}. \\ & = 1 \end{align}[/latex]
• [latex]\begin{align} \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& = 5{\left(r{s}^{2}\right)}^{2 - 2} && \text{Use the quotient rule}. \\ & = 5{\left(r{s}^{2}\right)}^{0} && \text{Simplify}. \\ & = 5\cdot 1 && \text{Use the zero exponent rule}. \\ & = 5 && \text{Simplify}. \end{align}[/latex]
• [latex]\dfrac{{t}^{7}}{{t}^{7}}[/latex]
• [latex]\dfrac{{\left(d{e}^{2}\right)}^{11}}{2{\left(d{e}^{2}\right)}^{11}}[/latex]
• [latex]\dfrac{{w}^{4}\cdot {w}^{2}}{{w}^{6}}[/latex]
• [latex]\dfrac{{t}^{3}\cdot {t}^{4}}{{t}^{2}\cdot {t}^{5}}[/latex]
• [latex]1[/latex]
• [latex]\dfrac{1}{2}[/latex]

Using the Negative Rule of Exponents

A general note: the negative rule of exponents, example: using the negative exponent rule.

• [latex]\dfrac{{\theta }^{3}}{{\theta }^{10}}[/latex]
• [latex]\dfrac{{z}^{2}\cdot z}{{z}^{4}}[/latex]
• [latex]\dfrac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}[/latex]
• [latex]\dfrac{{\theta }^{3}}{{\theta }^{10}}={\theta }^{3 - 10}={\theta }^{-7}=\dfrac{1}{{\theta }^{7}}[/latex]
• [latex]\dfrac{{z}^{2}\cdot z}{{z}^{4}}=\dfrac{{z}^{2+1}}{{z}^{4}}=\dfrac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\dfrac{1}{z}[/latex]
• [latex]\dfrac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}={\left(-5{t}^{3}\right)}^{4 - 8}={\left(-5{t}^{3}\right)}^{-4}=\dfrac{1}{{\left(-5{t}^{3}\right)}^{4}}[/latex]
• [latex]\dfrac{{\left(-3t\right)}^{2}}{{\left(-3t\right)}^{8}}[/latex]
• [latex]\dfrac{{f}^{47}}{{f}^{49}\cdot f}[/latex]
• [latex]\dfrac{2{k}^{4}}{5{k}^{7}}[/latex]
• [latex]\dfrac{1}{{\left(-3t\right)}^{6}}[/latex]
• [latex]\dfrac{1}{{f}^{3}}[/latex]
• [latex]\dfrac{2}{5{k}^{3}}[/latex]

Example: Using the Product and Quotient Rules

• [latex]{b}^{2}\cdot {b}^{-8}[/latex]
• [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}[/latex]
• [latex]\dfrac{-7z}{{\left(-7z\right)}^{5}}[/latex]
• [latex]{b}^{2}\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\frac{1}{{b}^{6}}[/latex]
• [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5 - 5}={\left(-x\right)}^{0}=1[/latex]
• [latex]\dfrac{-7z}{{\left(-7z\right)}^{5}}=\dfrac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1 - 5}={\left(-7z\right)}^{-4}=\dfrac{1}{{\left(-7z\right)}^{4}}[/latex]
• [latex]{t}^{-11}\cdot {t}^{6}[/latex]
• [latex]\dfrac{{25}^{12}}{{25}^{13}}[/latex]
• [latex]{t}^{-5}=\dfrac{1}{{t}^{5}}[/latex]
• [latex]\dfrac{1}{25}[/latex]

Finding the Power of a Product

A general note: the power of a product rule of exponents, example: using the power of a product rule.

• [latex]{\left(a{b}^{2}\right)}^{3}[/latex]
• [latex]{\left(2t\right)}^{15}[/latex]
• [latex]{\left(-2{w}^{3}\right)}^{3}[/latex]
• [latex]\dfrac{1}{{\left(-7z\right)}^{4}}[/latex]
• [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex]
• [latex]{\left(a{b}^{2}\right)}^{3}={\left(a\right)}^{3}\cdot {\left({b}^{2}\right)}^{3}={a}^{1\cdot 3}\cdot {b}^{2\cdot 3}={a}^{3}{b}^{6}[/latex]
• [latex]2{t}^{15}={\left(2\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}[/latex]
• [latex]{\left(-2{w}^{3}\right)}^{3}={\left(-2\right)}^{3}\cdot {\left({w}^{3}\right)}^{3}=-8\cdot {w}^{3\cdot 3}=-8{w}^{9}[/latex]
• [latex]\dfrac{1}{{\left(-7z\right)}^{4}}=\dfrac{1}{{\left(-7\right)}^{4}\cdot {\left(z\right)}^{4}}=\dfrac{1}{2,401{z}^{4}}[/latex]
• [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}={\left({e}^{-2}\right)}^{7}\cdot {\left({f}^{2}\right)}^{7}={e}^{-2\cdot 7}\cdot {f}^{2\cdot 7}={e}^{-14}{f}^{14}=\dfrac{{f}^{14}}{{e}^{14}}[/latex]
• [latex]{\left({g}^{2}{h}^{3}\right)}^{5}[/latex]
• [latex]{\left(5t\right)}^{3}[/latex]
• [latex]{\left(-3{y}^{5}\right)}^{3}[/latex]
• [latex]\dfrac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}[/latex]
• [latex]{\left({r}^{3}{s}^{-2}\right)}^{4}[/latex]
• [latex]{g}^{10}{h}^{15}[/latex]
• [latex]125{t}^{3}[/latex]
• [latex]-27{y}^{15}[/latex]
• [latex]\dfrac{1}{{a}^{18}{b}^{21}}[/latex]
• [latex]\dfrac{{r}^{12}}{{s}^{8}}[/latex]

Finding the Power of a Quotient

A general note: the power of a quotient rule of exponents, example: using the power of a quotient rule.

• [latex]{\left(\dfrac{4}{{z}^{11}}\right)}^{3}[/latex]
• [latex]{\left(\dfrac{p}{{q}^{3}}\right)}^{6}[/latex]
• [latex]{\left(\dfrac{-1}{{t}^{2}}\right)}^{27}[/latex]
• [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex]
• [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex]
• [latex]{\left(\dfrac{4}{{z}^{11}}\right)}^{3}=\dfrac{{\left(4\right)}^{3}}{{\left({z}^{11}\right)}^{3}}=\dfrac{64}{{z}^{11\cdot 3}}=\dfrac{64}{{z}^{33}}[/latex]
• [latex]{\left(\dfrac{p}{{q}^{3}}\right)}^{6}=\dfrac{{\left(p\right)}^{6}}{{\left({q}^{3}\right)}^{6}}=\dfrac{{p}^{1\cdot 6}}{{q}^{3\cdot 6}}=\dfrac{{p}^{6}}{{q}^{18}}[/latex]
• [latex]{\left(\dfrac{-1}{{t}^{2}}\right)}^{27}=\dfrac{{\left(-1\right)}^{27}}{{\left({t}^{2}\right)}^{27}}=\dfrac{-1}{{t}^{2\cdot 27}}=\dfrac{-1}{{t}^{54}}=-\dfrac{1}{{t}^{54}}[/latex]
• [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}={\left(\dfrac{{j}^{3}}{{k}^{2}}\right)}^{4}=\dfrac{{\left({j}^{3}\right)}^{4}}{{\left({k}^{2}\right)}^{4}}=\dfrac{{j}^{3\cdot 4}}{{k}^{2\cdot 4}}=\dfrac{{j}^{12}}{{k}^{8}}[/latex]
• [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}={\left(\dfrac{1}{{m}^{2}{n}^{2}}\right)}^{3}=\dfrac{{\left(1\right)}^{3}}{{\left({m}^{2}{n}^{2}\right)}^{3}}=\dfrac{1}{{\left({m}^{2}\right)}^{3}{\left({n}^{2}\right)}^{3}}=\dfrac{1}{{m}^{2\cdot 3}\cdot {n}^{2\cdot 3}}=\dfrac{1}{{m}^{6}{n}^{6}}[/latex]
• [latex]{\left(\dfrac{{b}^{5}}{c}\right)}^{3}[/latex]
• [latex]{\left(\dfrac{5}{{u}^{8}}\right)}^{4}[/latex]
• [latex]{\left(\dfrac{-1}{{w}^{3}}\right)}^{35}[/latex]
• [latex]{\left({p}^{-4}{q}^{3}\right)}^{8}[/latex]
• [latex]{\left({c}^{-5}{d}^{-3}\right)}^{4}[/latex]
• [latex]\dfrac{{b}^{15}}{{c}^{3}}[/latex]
• [latex]\dfrac{625}{{u}^{32}}[/latex]
• [latex]\dfrac{-1}{{w}^{105}}[/latex]
• [latex]\dfrac{{q}^{24}}{{p}^{32}}[/latex]
• [latex]\dfrac{1}{{c}^{20}{d}^{12}}[/latex]

Simplifying Exponential Expressions

Example: simplifying exponential expressions.

• [latex]{\left(6{m}^{2}{n}^{-1}\right)}^{3}[/latex]
• [latex]{17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}[/latex]
• [latex]{\left(\dfrac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}[/latex]
• [latex]\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)[/latex]
• [latex]{\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}[/latex]
• [latex]\dfrac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}[/latex]
• [latex]\begin{align} {\left(6{m}^{2}{n}^{-1}\right)}^{3}& = {\left(6\right)}^{3}{\left({m}^{2}\right)}^{3}{\left({n}^{-1}\right)}^{3}&& \text{The power of a product rule} \\ & = {6}^{3}{m}^{2\cdot 3}{n}^{-1\cdot 3}&& \text{The power rule} \\ & = 216{m}^{6}{n}^{-3}&& \text{Simplify}. \\ & = \frac{216{m}^{6}}{{n}^{3}}&& \text{The negative exponent rule} \end{align}[/latex]
• [latex]\begin{align} {17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}& =& {17}^{5 - 4-3}&& \text{The product rule} \\ & = {17}^{-2}&& \text{Simplify}. \\ & = \frac{1}{{17}^{2}}\text{ or }\frac{1}{289}&& \text{The negative exponent rule} \end{align}[/latex]
• [latex]\begin{align} {\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}& = \frac{{\left({u}^{-1}v\right)}^{2}}{{\left({v}^{-1}\right)}^{2}}&& \text{The power of a quotient rule} \\ & = \frac{{u}^{-2}{v}^{2}}{{v}^{-2}}&& \text{The power of a product rule} \\ & = {u}^{-2}{v}^{2-\left(-2\right)}&& \text{The quotient rule} \\ & = {u}^{-2}{v}^{4}&& \text{Simplify}. \\ & = \frac{{v}^{4}}{{u}^{2}}&& \text{The negative exponent rule} \end{align}[/latex]
• [latex]\begin{align} \left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)& =& -2\cdot 5\cdot {a}^{3}\cdot {a}^{-2}\cdot {b}^{-1}\cdot {b}^{2}&& \text{Commutative and associative laws of multiplication} \\ & = -10\cdot {a}^{3 - 2}\cdot {b}^{-1+2}&& \text{The product rule} \\ & = -10ab&& \text{Simplify}. \end{align}[/latex]
• [latex]\begin{align} {\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}& = {\left({x}^{2}\sqrt{2}\right)}^{4 - 4} && \text{The product rule} \\ & = {\left({x}^{2}\sqrt{2}\right)}^{0}&& \text{Simplify}. \\ & = 1&& \text{The zero exponent rule} \end{align}[/latex]
• [latex]\begin{align} \frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}& = \frac{{\left(3\right)}^{5}\cdot {\left({w}^{2}\right)}^{5}}{{\left(6\right)}^{2}\cdot {\left({w}^{-2}\right)}^{2}}&& \text{The power of a product rule} \\ & = \frac{{3}^{5}{w}^{2\cdot 5}}{{6}^{2}{w}^{-2\cdot 2}}&& \text{The power rule} \\ & = \frac{243{w}^{10}}{36{w}^{-4}} && \text{Simplify}. \\ & = \frac{27{w}^{10-\left(-4\right)}}{4}&& \text{The quotient rule and reduce fraction} \\ & = \frac{27{w}^{14}}{4}&& \text{Simplify}. \end{align}[/latex]
• [latex]{\left(2u{v}^{-2}\right)}^{-3}[/latex]
• [latex]{x}^{8}\cdot {x}^{-12}\cdot x[/latex]
• [latex]{\left(\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2}[/latex]
• [latex]\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right)[/latex]
• [latex]{\left(\frac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\frac{4}{9}t{w}^{-2}\right)}^{3}[/latex]
• [latex]\dfrac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}[/latex]
• [latex]\dfrac{{v}^{6}}{8{u}^{3}}[/latex]
• [latex]\dfrac{1}{{x}^{3}}[/latex]
• [latex]\dfrac{{e}^{4}}{{f}^{4}}[/latex]
• [latex]\dfrac{27r}{s}[/latex]
• [latex]\dfrac{16{h}^{10}}{49}[/latex]

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Exercise 60 Page   422

Begin by writing an expression that models a single year. Then, consider how that expression could change to account for the other time frames using exponents .

Check the answer

Practice exercises, asses your skill.

To find the budget from 2 years ago, let's create a model of the budget using some concrete examples.

Creating a Model Expression

Recall that the budget started as $ 500. The budget then doubles each year.

Compare the number of years with the exponents in the final expressions. Can you find a pattern? In each of the final expressions, there is an exponent whose value matches the number of years that have passed. Therefore, if y years have passed, we would get the following expression. 2^y* 500 Using this expression, we can find the budget for any number of years! Let's use this expression to find the budget for two years ago.

The Budget 2 Years Ago

a^(- m)=1/a^m

Calculate power

1/b* a = a/b

Calculate quotient

Common Core State Standards Initiative

High School: Algebra

Standards in this domain:, ccss.math.content.hsa.introduction introduction, expressions..

An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances.

Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05 p can be interpreted as the addition of a 5% tax to a price p . Rewriting p + 0.05 p as 1.05 p shows that adding a tax is the same as multiplying the price by a constant factor.

Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05 p is the sum of the simpler expressions p and 0.05 p . Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.

A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.

Equations and inequalities.

An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.

An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2 x + 1 = 0 is a rational number, not an integer; the solutions of x 2 - 2 = 0 are real numbers, not rational numbers; and the solutions of x 2 + 2 = 0 are complex numbers, not real numbers.

The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = (( b 1 + b 2 )/2) h , can be solved for h using the same deductive process. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.

Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.

Algebra Overview

• Seeing Structure in Expressions
• Interpret the structure of expressions
• Write expressions in equivalent forms to solve problems

Arithmetic with Polynomials and Rational Functions

• Perform arithmetic operations on polynomials
• Understand the relationship between zeros and factors of polynomials
• Use polynomial identities to solve problems
• Rewrite rational functions

Creating Equations

• Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities

• Understand solving equations as a process of reasoning and explain the reasoning
• Solve equations and inequalities in one variable
• Solve systems of equations
• Represent and solve equations and inequalities graphically

Mathematical Practices

• Make sense of problems and persevere in solving them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated reasoning.

Arithmetic with Polynomials & Rational Expressions

Creating equations ✭, reasoning with equations & inequalities.

• Standards for Mathematical Practice
• How to read the grade level standards
• Introduction
• Counting & Cardinality
• Operations & Algebraic Thinking
• Number & Operations in Base Ten
• Measurement & Data
• Number & Operations—Fractions¹
• Number & Operations in Base Ten¹
• Number & Operations—Fractions
• Ratios & Proportional Relationships
• The Number System
• Expressions & Equations
• Statistics & Probability
• The Real Number System
• Quantities*
• The Complex Number System
• Vector & Matrix Quantities
• Arithmetic with Polynomials & Rational Expressions
• Creating Equations*
• Reasoning with Equations & Inequalities
• Interpreting Functions
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