Algebra 1 Assignment Find Each Product Or Quotient

In earlier chapters we introduced powers.

$$x^{3}=x\cdot x\cdot x$$

There are a couple of operations you can do on powers and we will introduce them now.

We can multiply powers with the same base

$$x^{4}\cdot x^{2}=\left (x\cdot x\cdot x\cdot x \right )\cdot \left ( x\cdot x \right )=x^{6}$$

This is an example of the product of powers property tells us that when you multiply powers with the same base you just have to add the exponents.

$$x^{a}\cdot x^{b}=x^{a+b}$$

We can raise a power to a power

$$\left ( x^{2} \right )^{4}= \left (x\cdot x \right )\cdot \left (x\cdot x \right ) \cdot \left ( x\cdot x \right )\cdot \left ( x\cdot x \right )=x^{8}$$

This is called the power of a power property and says that to find a power of a power you just have to multiply the exponents.

When you raise a product to a power you raise each factor with a power

$$\left (xy \right )^{2}= \left ( xy \right )\cdot \left ( xy \right )= \left ( x\cdot x \right )\cdot \left ( y\cdot y \right )=x^{2}y^{2}$$

This is called the power of a product property

$$\left (xy \right )^{a}= x^{a}y^{a}$$

As well as we could multiply powers we can divide powers.

$$\frac{x^{4}}{x^{2}}=\frac{x\cdot x\cdot {\color{red} \not}{x}\cdot {\color{red} \not}{x}}{{\color{red} \not}{x}\cdot {\color{red} \not}{x}}=x^{2}$$

This is an example of the quotient of powers property and tells us that when you divide powers with the same base you just have to subtract the exponents.

$$\frac{x^{a}}{x^{b}}=x^{a-b},\: \: x\neq 0$$

When you raise a quotient to a power you raise both the numerator and the denominator to the power.

$$\left (\frac{x}{y} \right )^{2}=\frac{x}{y}\cdot \frac{x}{y}=\frac{x\cdot x}{y\cdot y}=\frac{x^{2}}{y^{2}}$$

This is called the power of a quotient power

$$\left (\frac{x}{y} \right )^{a}=\frac{x^{a}}{y^{a}},\: \: y\neq 0$$

When you raise a number to a zero power you'll always get 1.

$$1=\frac{x^{a}}{x^{a}}=x^{a-a}=x^{0}$$

$$x^{0}=1,\: \: x\neq 0$$

Negative exponents are the reciprocals of the positive exponents.

$$x^{-a}=\frac{1}{x^{a}},\: \: x\neq 0$$

$$x^{a}=\frac{1}{x^{-a}},\: \: x\neq 0$$

The same properties of exponents apply for both positive and negative exponents.

In earlier chapters we talked about the square root as well. The square root of a number x is the same as x raised to the 0.5th power

$$\sqrt{x}=\sqrt[2]{x}=x^{\frac{1}{2}}$$


Video lesson

Simplify the following expression using the properties of exponents

$$\frac{( 7^{5}) ^{10}\cdot 7^{200}}{\left ( 7^{-2} \right )^{30}}$$

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Common Core Algebra 1

These interactive lessons use dynamic graphing and guided discovery to strengthen and connect symbolic and visual reasoning. They give the student a hands-on visual exposition of all Common Core Algebra 1 topics, reinforced by adaptive exercises and randomly generated tests. All exercises and tests are checked and graded automatically. Hover the mouse over a link below to see an example from that lesson, or click on a test link to see a concise summary of a group of lessons. Relevant standards are listed after each lesson’s exercises, with a ' (prime) denoting a distinct link. We also provide a course glossary.

The Common Core and other standards cover many algebraic topics before the first actual High School Algebra course. Therefore, students may have already completed many of the lessons and exercises below in an earlier grade, or in a summer preparatory course. If so, those assignments will be treated as already completed in this course also.

Students should have scratch paper available, access to (software or hardware) numeric calculators except during the Arithmetic Review chapter, and other students or a teacher to ask for help when they are stuck. All students are encouraged to both give and receive mathematical explanations with their peers. Please send us your comments, questions and suggestions.

Arithmetic Review (Optional)

  1. Arithmetic on a Grid: Counting unit squares. Sum, difference, product, quotient.
  2. Two-Digit Addition: Two-digit numbers shown as $a(10)+b$ on grid. Addition with regrouping, carrying.
    Exercises: addition
  3. Two-Digit Subtraction and Multiplication: Subtraction with borrowing, multiplication with regrouping.
    Exercises: subtraction, multiplication, division
  4. Negative Numbers: Green squares have value $+1$, pink squares have value $-1$. Sums and differences. Word problems involving debt.
    Exercises: addition, subtraction
  5. Multiplication and Division with Negative Numbers: Products and quotients with negative numbers on the grid, introduced through analogy and pattern-matching.
    Exercises: multiplication, division
  6. Fraction Addition and Subtraction: Visualizing fractions using pie slices. Reducing to lowest terms. Finding common denominators.
    Exercises: addition, subtraction
  7. Fraction Multiplication and Division: Using an expanded grid to visualize fractions.
    Exercises: multiplication, division
  8. Arithmetic Test

Variables, Expressions, and Simple Equations

  1. Variables and Expressions: Expressions as quantities. Simple linear expressions. Sliders for variables.
    Exercises: evaluation6.EE.2
  2. Word Problems: Cost of $n$ items. Multi-variable expressions.
    Exercises: expressions6.EE.6, A.SSE.1a, A.CED.2
  3. More Complicated Expressions: Examples involving negative numbers, division, fractions.
    Exercises: evaluation, divisionA.SSE.1a
  4. Equations as Sentences: Solving equations by trial and error, or by sliding a slider. Inverse problems as word problems.
    Exercises: solving6.EE.5, A.CED.1
  5. Grouping in Addition and Subtraction Problems: Parentheses, associative law of addition, order of operations for addition and subtraction. Simplifying expressions using the associativity of addition to regroup, including for subtraction.
    Exercises: evaluation, simplifying6.EE.3, A.APR.1
  6. Solving $x+b=c$: Adding a constant to both sides.
    Exercises: solving6.EE.7, A.REI.1, A.REI.3
  7. Grouping in Multiplication Problems: Order of operations including multiplication. Multiplying three numbers, associative law of multiplication and its use in simplifying expressions.
    Exercises: evaluation 1, evaluation 2, multiplication, division6.EE.3, A.APR.1
  8. Solving $ax+b=c$: Adding $-b$ and then multiplying by $$1/a$$, or dividing by $a$.
    Exercises: solving 1, solving 26.EE.7, A.REI.1, A.REI.3
  9. Applications of Linear Equations: Balanced scale, budgeting, temperature conversion.
    Exercises: solving6.EE.7, 7.EE.4a, A.CED.1, F.BF.1a
  10. Variables, Expressions, and Simple Equations Test 1
  11. The Distributive Law and Combining Like Terms: Expanding $a(x+c)$, $a(bx+c)$, $-(bx+c)$. Simplifying $ax+bx$ to $[a+b]x$.
    Exercises: simplifying 1, simplifying 2, combining like terms6.EE.3, A.APR.1
  12. Manipulating Linear Expressions: Adding, subtracting, scaling, simplifying. Commutativity of addition and multiplication. Zero and one laws.
    Exercises: simplifying 1, simplifying 26.EE.3, 7.EE.1, A.APR.1
  13. Solving $ax+b=cx+d$: Adding $-cx-b$ to both sides.
    Exercises: solving8.EE.7b, A.REI.1, A.REI.3
  14. Variables, Expressions, and Simple Equations Test 2

Linear Graphs

  1. Points and Coordinates: Connecting the ordered pair $(x, y)$ with the coordinate plane. Quadrants.
    Exercises: quadrants, coordinates6.NS.6bc
  2. Investigating $y=x+b$: Finding the $y$-intercept $(0, b)$ of a line.
    Exercises: $y$-interceptsF.IF.7a, F.BF.3
  3. Investigating $y=mx$: Lines with slope $m$, positive or negative.
    Exercises: slopes8.EE.5, 8.F.2, F.LE.1a
  4. Investigating $y=mx+b$: Slope and $y$-intercept.
    Exercises: $y$-intercepts 1, slopes 1, $y$-intercepts 2, slopes 2, equations8.F.3, F.IF.7a, F.LE.1a
  5. Solving Equations Graphically: Graphing each side and finding the intersection, e.g. for $ax+b=cx+d$.
    Exercises: solvingA.REI.11
  6. Finding Formulas for Approximately Linear Data: Simple statistics application. Uses root-mean-square error informally. 8.SP.2, F.IF.4, S.ID.6ac
  7. Slopes, Rates of Change, and Similar Triangles: Calculating the slope of a line from any two points on it. Derivation of the equation $y=mx$ or $y=mx+b$ from a line’s slope and $y$-intercept.
    Exercises: computing slopes8.F.4, F.LE.1b, F.LE.2
  8. Finding Equations for Lines: Point-slope form.
    Exercises: point-slope formF.LE.2
  9. Parallel and Perpendicular Lines: Slopes $m$ and $$-1/m$$.
    Exercises: parallel lines, perpendicular lines8.EE.8c
  10. Equations for Lines in Standard Form: $Ax + By = C$. Converting to slope-intercept form. Computing slopes, $x$- and $y$-intercepts.
    Exercises: converting to slope-intercept form, $y$-intercepts, $x$-interceptsA.CED.4, A.REI.10
  11. Linear Graphs Test

Inequalities, Absolute Value, and Square Roots

  1. Linear Inequalities: Half-plane solution graphs, with or without boundary line.
    Exercises: evaluating, identifying inequalitiesA.CED.1, A.REI.12
  2. Equivalent Inequalities: Adding a constant to both sides of an inequality. Multiplying both sides by a positive or negative number.
    Exercises: addition, multiplication
  3. Solving Inequalities Algebraically: Finding algebraic solutions to $ax+b < c$ and similar inequalities.
    Exercises: solvingA.REI.3
  4. Absolute Value: Definition of absolute value. Graphing and solving equations which include absolute values. $|ab| = |a||b|$.
    Exercises: solving6.NS.7c, F.IF.7b'
  5. Square Roots: Definition and computation of square roots. Simplifying square roots via product, quotient, and absolute value identities.
    Exercises: simplifying8.EE.2, A.REI.4b
  6. Rational and Irrational Numbers: Definitions. If $q$ and $r$ are rational, then so are $q+r$, $q-r$, $qr$, and $$q/r$$ if $r ≠ 0$; but $√2$ for instance is not.
    Exercises: repeating decimals, rationality8.NS.1, 8.EE.2, N.RN.3
  7. Inequalities, Absolute Value, and Square Roots Test

Descriptive Statistics

  1. Summarizing a Collection of Measurements: Converting to common units. Dot (line) plots. Median and mean. Histograms.
    Exercises: median from a list, median from frequencies, mean6.SP.3, 6.SP.4, N.Q.1, N.Q.2, N.Q.3, S.ID.1, S.ID.2
  2. Box Plots: First and third quartiles, interquartile range. Comparing data collections.
    Exercises: quartiles, box plots6.SP.4, S.ID.1, S.ID.2, S.ID.3
  3. Measuring Spread in Data: Mean absolute deviation, standard deviation.
    Exercises: mean absolute deviations, standard deviations6.SP.3, S.ID.2
  4. Analyzing Outliers and Choosing Statistics: Important and unimportant outliers. Choosing the best statistics to describe a collection of data values. 6.SP.5, S.ID.3
  5. Data Relating Two Categorizations: Two-way frequency tables. Joint, marginal, and conditional relative frequencies. Possible associations and trends in the data.
    Exercises: joint frequencies, marginal frequencies, conditional relative frequencies8.SP.4, S.ID.5
  6. Data Relating Two Measurements: Scatter plots. Linear model, residuals, root-mean-square error, interpretation of slope and $y$-intercept. Correlation coefficient, correlation is not causation.
    Exercises: root mean square errors, correlation coefficients8.SP.1, 8.SP.3, S.ID.6abc, S.ID.7, S.ID.8, S.ID.9
  7. Descriptive Statistics Test

Systems of Linear Equations or Inequalities

  1. Solving Systems of Linear Equations by Graphing: Solutions are intersections of graphs.
    Exercises: checking, solving8.EE.8a, A.REI.6
  2. Solving Systems of Linear Equations by Subtraction: Solving for a difference equal to 0.
    Exercises: solving8.EE.8b, A.REI.5, A.REI.6
  3. Solving Systems of Linear Equations by Multiplication and Addition: Eliminating a variable in an equation.
    Exercises: solving8.EE.8b, A.REI.5, A.REI.6
  4. Applications of Systems of Linear Equations: Cost, revenue, profit. Birth and death rates. 8.EE.8c, 8.SP.3, A.CED.2, F.LE.5
  5. Mixture Problems: Mathematics of mixing different concentrations.
    Exercises: solvingA.CED.2, A.REI.6
  6. Systems of Linear Inequalities: Solution graphs are intersections of half-planes.
    Exercises: solvingA.REI.12
  7. Linear Optimization: Maximizing profit subject to linear inequalities. A.CED.3
  8. Systems of Linear Equations or Inequalities Test

Quadratic Expressions, Graphs, and Equations

  1. Investigating $y=(x-h)^2+k$: Vertex, axis of symmetry.
    Exercises: vertex, axis of symmetry8.F.3, F.IF.7a', F.IF.9'
  2. Investigating $y=a(x-h)^2+k$: Positive and negative $a$. Roots.
    Exercises: vertex, axis of symmetry, number of rootsF.IF.7a'
  3. Quadratic Polynomials: Monomials and polynomials. Adding, subtracting, scaling quadratics. Multiplying two monic linear factors to obtain a quadratic polynomial. Uses the algebra grid.
    Exercises: addition, subtraction, multiplication 1, multiplication 2A.APR.1
  4. More Quadratic Polynomials: Multiplying non-monic linear polynomials. Performing several simplifications on one quadratic expression.
    Exercises: simplifyingA.APR.1
  5. Factoring $x^2+bx+c$: Graphing $y=(x-r)(x-s)$. Solving $x^2+bx+c=0$ by factoring.
    Exercises: factoringA.SSE.3a, A.REI.4b
  6. Factoring $ax^2+bx+c$: Constant and linear factors. Clearing denominators. Moving all terms to one side of an equation.
    Exercises: factoring 1, factoring 2, clearing denominators, solvingA.SSE.2, A.REI.4b
  7. Quadratics Test 1
  8. Investigating $y=ax^2+bx$: Roots, axis of symmetry, vertex.
    Exercises: axis of symmetry, vertexF.IF.8a'
  9. Investigating $y=ax^2+bx+c$: Axis of symmetry, vertex, rewriting in vertex form.
    Exercises: axis of symmetry, vertexF.IF.8a'
  10. Solving Quadratic Equations: Completing the square, solving any quadratic equation.
    Exercises: solving, converting vertex form to standard formA.SSE.3b, A.REI.4ab, F.IF.8a'
  11. The Quadratic Formula: Solving any quadratic equation, using a formula.
    Exercises: discriminants, solving 1, solving 2, solving 3A.REI.4ab
  12. Quadratics Test 2
  13. Applications of Quadratic Equations: Objects in free fall, the shape of a suspension bridge, profit maximization.
    Exercises: solving8.F.5, A.SSE.1a', A.CED.1', F.IF.4', F.BF.1a'
  14. Quadratic Systems of Equations: Using substitution to eliminate a variable.
    Exercises: linear systems, number of solutions, solving 1, solving 2A.REI.7
  15. Quadratics Test 3

Exponential Growth and Decay

  1. The Laws of Exponents: For all positive integers $c$ and $d$, $a^c a^d = a^{c+d}$, $(ab)^d = a^d b^d$, and $(a^c)^d = a^{cd}$. If $a ≠ 0$ and $b ≠ 0$, we can extend these laws to all integers $c$ and $d$ using $a^0 = 1$ and $$a^{- d} = 1/a^d$$.
    Exercises: simplifying 1, simplifying 28.EE.1, F.LE.1a
  2. Rational Exponents: Given $a > 0$, and integers $m$ and $n$ with $n > 0$, we define $$a^{m∕n} = √^n{a^m}$$. Then the laws of exponents still hold for positive bases and rational exponents. Graph of $y=a^x$ for various $a > 0$.
    Exercises: $n$th roots, simplifyingN.RN.1, N.RN.2, F.IF.6, F.IF.7e, F.IF.8b', F.LE.2
  3. Comparing Exponential Graphs: Graphs of $y = m a^x$ and $y = a^{x+c}$, which coincide when $a^c = m$. Graphs of $y = b^x$ and $y = a^{rx}$, which coincide when $a^r = b$. Compare linear, quadratic, and exponential growth, including for large $x$.
    Exercises: rewriting 1, rewriting 2, comparing growth ratesF.IF.8b', F.BF.3, F.LE.3'
  4. Applications of Exponential Growth: Compound interest, musical pitches, biological growth.
    Exercises: compound interestA.SSE.1ab', A.SSE.3c, A.CED.2', F.LE.1c, F.LE.5
  5. Applications of Exponential Decay: Carbon dating, cooling objects. A.CED.1', A.REI.11, F.IF.4, F.BF.1b, F.LE.1c, F.LE.5
  6. Exponentials Test

Functions

  1. Sets, Relations, and Functions: Sets, relations, domain, range. When is a relation a function?
    Exercises: relations from tables, relations from graphs8.F.1, F.IF.1
  2. Functions as Transformations: Function notation $f(x)$. Defining a function by a table, formula, or graph. Composition.
    Exercises: evaluationF.IF.1, F.IF.2
  3. Inverse Functions: Definition and computation of the inverse of a function. One-to-one functions. Square and cube roots. The inverse of a linear function.
    Exercises: one-to-one functions, inverting linear functionsF.IF.7b', F.BF.4
  4. Shifting Functions: Moving a graph through composition of functions. Periodic functions.
    Exercises: shifting functionsF.BF.3'
  5. Stretching and Flipping Functions: Dilating or reflecting a graph through composition of functions. Even and odd functions and polynomials.
    Exercises: stretching graphs, even and odd functionsF.BF.3'
  6. Piecewise-Defined Functions: Piecewise-defined functions, step functions, and the floor and ceiling functions.
    Exercises: evaluationF.IF.5', F.IF.7b'
  7. Sequences: Arithmetic, geometric, and Fibonacci sequences. The sum $1+2+...+n$.
    Exercises: arithmetic sequences, geometric sequencesF.IF.3, F.IF.5, F.BF.2, F.LE.2
  8. Applications of Functions: Temperature conversion, profit, rice on a chessboard. F.IF.2, F.IF.5, F.BF.1a, F.BF.3
  9. Functions Test

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