Glencoe Algebra 1: Multiplication and Division Properties of Exponents, Rational Exponents, and Exponential Functions Study Guide

Lesson 7-1: Multiplication Properties of Exponents

• Learning Objectives and Goals:

  • Multiply monomials using the Product of Powers property.

  • Simplify complex expressions involving monomials using Power of a Power and Power of a Product properties.

  • Perform operations on expressions with exponents.

• New Vocabulary Definitions:

  • Monomial: A number, a variable, or a product of a number and one or more variables with nonnegative integer exponents. It has only one term.

  • Constant: A monomial that is a real number (e.g., $\frac{3}{4}$).

• Identifying Monomials:

  • Monomial Example: $17 - c$ is NOT a monomial because the expression involves subtraction, meaning it has more than one term.

  • Monomial Example: $8f^2g$ IS a monomial because it is the product of a number and two variables.

  • Monomial Example: $\frac{3}{4}$ IS a monomial because it is a constant.

  • Monomial Example: $\frac{5}{t}$ is NOT a monomial because the expression involves division by a variable.

• Key Concept: Product of Powers:

  • Word Definition: To multiply two powers that have the same base, add their exponents.

  • Mathematical Symbols: For any real number $a$ and any integers $m$ and $p$, $a^m \cdot a^p = a^{m+p}$.

  • Example 1: $b^3 \cdot b^5 = b^{3+5}$ or $b^8$.

  • Example 2: $g^4 \cdot g^6 = g^{4+6}$ or $g^{10}$.

• Product of Powers Examples:

  • Simplifying $(r^4)(-12r^7)$.

    • Group coefficients and variables: $[1 \cdot (-12)](r^4)(r^7)$.

    • Apply Product of Powers: $[1 \cdot (-12)](r^{4+7})$.

    • Final Answer: $-12r^{11}$.

  • Simplifying $(6cd^5)(5c^5d^2)$.

    • Group coefficients and variables: $(6 \cdot 5)(c \cdot c^5)(d^5 \cdot d^2)$.

    • Apply Product of Powers: $(6 \cdot 5)(c^{1+5})(d^{5+2})$.

    • Final Answer: $30c^6d^7$.

• Key Concept: Power of a Power:

  • Word Definition: To find the power of a power, multiply the exponents.

  • Mathematical Symbols: For any real number $a$ and any integers $m$ and $p$, $(a^m)^p = a^{m \cdot p}$.

  • Example 1: $(6^3)^5 = 6^{3 \cdot 5}$ or $6^{15}$.

  • Example 2: $(9^6)^7 = 9^{6 \cdot 7}$ or $9^{42}$.

• Power of a Power Simplification:

  • Simplifying $[(2^3)^3]^2$.

    • Inner power: $(2^3)^3 = 2^{3 \cdot 3} = 2^9$.

    • Outer power: $(2^9)^2 = 2^{9 \cdot 2} = 2^{18}$.

    • Final Answer: $2^{18}$ or $262,144$.

• Key Concept: Power of a Product:

  • Word Definition: To find the power of a product, find the power of each factor and multiply.

  • Mathematical Symbols: For any real numbers $a$ and $b$ and any integer $m$, $(ab)^m = a^m b^m$.

  • Example 1: $(-2xy^3)^5 = (-2)^5 x^5 (y^3)^5$ or $-32x^5y^{15}$.

• Application in Geometry:

  • Finding the Volume of a Cube with side length $5xyz$.

    • Formula for volume of a cube: $V = s^3$.

    • Substitution: $V = (5xyz)^3$.

    • Power of a Product: $5^3 x^3 y^3 z^3$.

    • Final Answer: $125x^3y^3z^3$.

• Criteria for a Simplified Monomial Expression:

  • Each variable base appears exactly once.

  • There are no powers of powers.

  • All fractions are in simplest form.

• Exhaustive Simplification Example:

  • Simplifying $[(8g^3h^4)^2]^2(2gh^5)^4$.

    • Power of a Power (first term): $(8g^3h^4)^4(2gh^5)^4$.

    • Power of a Product: $(8)^4(g^3)^4(h^4)^4 (2)^4 g^4 (h^5)^4$.

    • Evaluate and apply Power of a Power: $4096g^{12}h^{16}(16)g^4h^{20}$.

    • Commutative Property: $4096(16) g^{12} \cdot g^4 \cdot h^{16} \cdot h^{20}$.

    • Product of Powers: $65,536g^{16}h^{36}$.

Lesson 7-2: Division Properties of Exponents

• Learning Objectives and Goals:

  • Find the quotient of two monomials.

  • Simplify expressions containing negative and zero exponents.

• New Vocabulary Definitions:

  • Zero Exponent: Any nonzero number raised to the power of zero.

  • Negative Exponent: For any nonzero number $a$ and integer $n$, $a^{-n}$ is the reciprocal of $a^n$.

  • Order of Magnitude: Used to compare large or small numbers by powers of 10.

• Key Concept: Quotient of Powers:

  • Word Definition: To divide two powers with the same base, subtract the exponents.

  • Mathematical Symbols: For any nonzero number $a$, and any integers $m$ and $p$, $\frac{a^m}{a^p} = a^{m-p}$.

  • Example 1: $\frac{c^{11}}{c^8} = c^{11-8}$ or $c^3$.

  • Example 2: $\frac{k^{14}}{k^2} = k^{12}$.

• Quotient of Powers Example:

  • Simplifying $\frac{x^3y^{12}}{x^2y^3}$.

    • Group same bases: $(\frac{x^3}{x^2})(\frac{y^{12}}{y^3})$.

    • Product of Powers (Subtraction): $x^{3-2}y^{12-3} = xy^9$.

• Key Concept: Power of a Quotient:

  • Word Definition: To find the power of a quotient, find the power of the numerator and the power of the denominator.

  • Mathematical Symbols: For any real numbers $a$ and $b \neq 0$, and any integer $m$, $(\frac{a}{b})^m = \frac{a^m}{b^m}$.

  • Example 1: $(\frac{3}{5})^4 = \frac{3^4}{5^4}$.

• Power of a Quotient Example:

  • Simplifying $(\frac{4c^3d^2}{5})^3$.

    • Apply Power of a Quotient: $\frac{(4c^3d^2)^3}{5^3}$.

    • Apply Power of a Product (Numerator): $\frac{4^3(c^3)^3(d^2)^3}{125}$.

    • Final Answer: $\frac{64c^9d^6}{125}$.

• Key Concept: Zero Exponent Property:

  • Word Definition: Any nonzero number raised to the zero power is equal to 1.

  • Mathematical Symbols: For any nonzero number $a$, $a^0 = 1$.

  • Examples: $15^0 = 1$; $(\frac{b}{20})^0 = 1$.

• Zero Exponent Example:

  • Simplifying $\frac{12m^8n^7}{8m^5n^{10}}^0$.

    • Since the entire bracket is raised to the 0 power, the answer is $1$.

  • Simplifying $\frac{m^0n^3}{n^2}$.

    • Substitute $m^0 = 1$: $\frac{1 \cdot n^3}{n^2}$.

    • Subtract exponents ($3-2$): $n^1$ or $n$.

• Key Concept: Negative Exponent Property:

  • Word Definition: For any nonzero number $a$ and any integer $n$, $a^{-n}$ is the reciprocal of $a^n$. Also, the reciprocal of $a^{-n}$ is $a^n$.

  • Mathematical Symbols: For any nonzero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$.

  • Example 1: $2^{-4} = \frac{1}{2^4}$ or $\frac{1}{16}$.

• Negative Exponent Example:

  • Simplifying $\frac{x^{-2}y^3z}{x}$.

    • Regroup: $(\frac{x^{-2}}{x})(y^3)(z)$.

    • Combine exponents: $x^{-2-1}y^3z = x^{-3}y^3z$.

    • Remove negatives: $\frac{y^3z}{x^3}$.

  • Complex Negative Exponent: $\frac{5p^{-2}}{q^{-1}r^{-3}}$.

    • Final Answer by moving negative powers to the opposite side of the fraction: $\frac{5qr^3}{p^2}$.

• Real World Example: Order of Magnitude:

  • Scenario: Darin has $\$123,456$ in savings. Tabo has $\$156$.

  • Standard Calculation: Darin is close to $\$100,000$ ($10^5$). Tabo is close to $\$100$ ($10^2$).

  • Ratio Calculation: $\frac{10^5}{10^2} = 10^3$.

  • Conclusion: Darin has about 1000 times ($10^3$) as much as Tabo, meaning Darin has 3 orders of magnitude as much money.

Lesson 7-3: Rational Exponents

• Learning Objectives and Goals:

  • Evaluate and rewrite expressions involving rational exponents.

  • Solve equations involving expressions with rational exponents.

• New Vocabulary Definitions:

  • Rational Exponent: An exponent that is a fraction.

  • Cube Root: If $a^3 = b$, then $a$ is the cube root of $b$.

  • nth Root: For any real numbers $a$ and $b$ and any positive integer $n$, if $a^n = b$, then $a$ is an nth root of $b$.

  • Exponential Equation: An equation in which variables occur as exponents.

• Key Concept: $b^{\frac{1}{2}}$:

  • Word Definition: For any nonnegative real number $b$, $b^{\frac{1}{2}} = \sqrt{b}$.

  • Examples: $16^{\frac{1}{2}} = \sqrt{16}$ or $4$.

• Key Concept: nth Root:

  • Word Definition: If $a^n = b$, then $\sqrt[n]{b} = a$.

  • Examples: $\sqrt[4]{256}$. Since $4 \cdot 4 \cdot 4 \cdot 4 = 256$, $\sqrt[4]{256} = 4$.

  • Examples: $\sqrt[6]{15,625}$. Since $5^6 = 15,625$, the answer is $5$.

• Key Concept: $b^{\frac{1}{n}}$:

  • Word Definition: For any positive real number $b$ and any integer $n > 1$, $b^{\frac{1}{n}} = \sqrt[n]{b}$.

  • Example: $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$.

  • Example: $1331^{\frac{1}{3}} = \sqrt[3]{1331} = 11$.

• Key Concept: $b^{\frac{m}{n}}$:

  • Power Formulation: For any positive real number $b$ and any integers $m$ and $n > 1$, $b^{\frac{m}{n}} = (\sqrt[n]{b})^m$ or $\sqrt[n]{b^m}$.

  • Example: $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2$ or $4$.

  • Evaluation Example: $32^{\frac{2}{5}} = (\sqrt[5]{32})^2 = 2^2$ or $4$.

  • Evaluation Example: $81^{\frac{5}{2}} = (\sqrt{81})^5 = 9^5$ or $59,049$.

• Key Concept: Power Property of Equality:

  • Word Definition: For any real number $b > 0$ and $b \neq 1$, $b^x = b^y$ if and only if $x = y$.

  • Example: If $5^x = 5^3$, then $x = 3$.

• Solving Exponential Equations:

  • Basic Example: Solve $9^x = 729$.

    • Rewrite $729$ as a power of 9: $9^x = 9^3$.

    • By Power Property of Equality: $x = 3$.

  • Intermediate Example: Solve $16^{2x-1} = 8$.

    • Rewrite with common base 2: $(2^4)^{2x-1} = 2^3$.

    • Power of a Power: $2^{8x-4} = 2^3$.

    • Equate exponents: $8x - 4 = 3$.

    • Solve: $8x = 7 \rightarrow x = \frac{7}{8}$.

• Real-World Application: Biology Populations:

  • Formula: $p = 40(2)^{\frac{t}{8}}$, where $p$ is population and $t$ is time in hours.

  • Problem: Find $t$ if $p = 20,480$.

  • Step 1: $20,480 = 40(2)^{\frac{t}{8}}$.

  • Step 2: Divide by 40: $512 = 2^{\frac{t}{8}}$.

  • Step 3: Rewrite 512 as $2^9$: $2^9 = 2^{\frac{t}{8}}$.

  • Step 4: Equate exponents: $9 = \frac{t}{8}$.

  • Result: $t = 72$ hours.

Lesson 7-4: Scientific Notation

• Learning Objectives and Goals:

  • Express numbers in scientific notation.

  • Find products and quotients of numbers expressed in scientific notation.

• New Vocabulary Definitions:

  • Scientific Notation: A way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is written in the form $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.

• Standard Form to Scientific Notation Procedure:

  • Step 1: Move the decimal point until it is to the right of the first nonzero digit. This results in the number $a$.

  • Step 2: Note the number of places $n$ and the direction moved.

  • Step 3: If moved left, $n$ is positive ($a \times 10^n$). If moved right, $n$ is negative ($a \times 10^{-n}$).

  • Step 4: Remove unnecessary trailing or leading zeros.

  • Example A (Large Number): $4,062,000,000,000 \rightarrow 4.062 \times 10^{12}$.

  • Example B (Small Number): $0.000000823 \rightarrow 8.23 \times 10^{-7}$.

• Scientific Notation to Standard Form Procedure:

  • Step 1: Note whether $n > 0$ (positive) or $n < 0$ (negative).

  • Step 2: If $n > 0$, move the decimal point $n$ places right. If $n < 0$, move it $-n$ places left.

  • Step 3: Insert placeholder zeros and commas.

  • Example A: $6.49 \times 10^5 = 649,000$.

  • Example B: $1.8 \times 10^{-3} = 0.0018$.

• Multiplication with Scientific Notation:

  • Example: Evaluate $(5 \times 10^{-6})(2.3 \times 10^{12})$.

    • Group coefficients and group powers: $(5 \times 2.3) \times (10^{-6} \times 10^{12})$.

    • Multiply: $11.5 \times 10^6$.

    • Adjust to proper scientific notation ($1.15 \times 10^1 \times 10^6$): $1.15 \times 10^7$.

    • Standard form: $11,500,000$.

• Division with Scientific Notation:

  • Example: Evaluate $\frac{4.5 \times 10^{-2}}{1.5 \times 10^{-3}}$.

    • Quotient of coefficients: $\frac{4.5}{1.5} = 3$.

    • Quotient of powers: $10^{-2 - (-3)} = 10^1$.

    • Result: $3 \times 10^1 = 30$.

• Real World Example: Business Revenue:

  • Newspaper circulation: $150$ thousand = $150,000$ or $1.5 \times 10^5$.

  • Advertising revenue: $\$1.2$ million = $\$1,200,000$ or $\$1.2 \times 10^6$.

Lesson 7-5: Exponential Functions

• Learning Objectives and Goals:

  • Graph exponential functions.

  • Identify data that display exponential behavior.

• New Vocabulary Definitions:

  • Exponential Function: A function that can be described by an equation of the form $y = ab^x$, where $a \neq 0$, $b > 0$, and $b \neq 1$.

  • Exponential Growth Function: A function where $a > 0$ and $b > 1$.

  • Exponential Decay Function: A function where $a > 0$ and $0 < b < 1$.

• Characteristics of Exponential Growth Graphs ($f(x) = ab^x, b > 1$):

  • Equation: $f(x) = ab^x$.

  • Domain: All real numbers.

  • Range: All positive real numbers.

  • Intercepts: One y-intercept (at $a$), no x-intercepts.

  • End Behavior: As $x$ increases, $f(x)$ increases; as $x$ decreases, $f(x)$ approaches $0$.

• Characteristics of Exponential Decay Graphs ($f(x) = ab^x, 0 < b < 1$):

  • Equation: $f(x) = ab^x$.

  • Domain: All real numbers.

  • Range: All positive real numbers (transcript incorrectly notes range as all negative reals, but the graph confirms all positive reals).

  • Intercepts: One y-intercept, no x-intercepts.

  • End Behavior: As $x$ increases, $f(x)$ approaches $0$; as $x$ decreases, $f(x)$ increases.

• Real-World Application: Car Depreciation:

  • Formula: $V = 25,000 \cdot 0.82^t$.

  • $V$ is value, $t$ is time in years.

  • Initial cost: $\$25,000$.

  • Question: Value after 5 years?

  • Calculation: $V = 25,000 \cdot 0.82^5 \approx 9268$.

  • Meaningful Values: $t \ge 0$ and $0 \le V \le 25,000$.

• Identifying Exponential Behavior from Data:

  • Look for a pattern in the domain and range.

  • If domain values are at regular intervals and range values have a common factor (not common difference), the behavior is exponential.

  • Example Data: $(0, 10), (10, 25), (20, 62.5)$.

    • Common factor in range: $\frac{25}{10} = 2.5$; $\frac{62.5}{25} = 2.5$.

    • This indicates exponential behavior: $y = a(2.5)^x$.

Lesson 7-6: Growth and Decay

• Learning Objectives and Goals:

  • Solve problems involving exponential growth (e.g., populations, investments).

  • Solve problems involving exponential decay (e.g., charity trends, depreciation).

• New Vocabulary Definitions:

  • Compound Interest: Interest earned or paid on both the initial principal and previously earned interest.

• Key Concept: Equation for Exponential Growth:

  • Equation: $y = a(1 + r)^t$.

  • $a$: Initial amount.

  • $y$: Final amount.

  • $t$: Time.

  • $r$: Rate of growth as a decimal ($r > 0$).

• Exponential Growth Application: Population:

  • Problem: Town population was $280,000$ in 2015; growth rate is $0.85\%$.

  • Equation: $y = 280,000(1 + 0.0085)^t$ or $y = 280,000(1.0085)^t$.

  • To find population in 2028: Set $t = 13$ ($2028 - 2015$).

  • Calculation: $y = 280,000(1.0085)^{13} \approx 312,216$ (Note: page 170 uses a 2018-2028 window for $t=10$, resulting in $304,731$).

• Key Concept: Equation for Compound Interest:

  • Equation: $A = P(1 + \frac{r}{n})^{nt}$.

  • $A$: Current amount.

  • $P$: Principal (initial amount).

  • $r$: Annual interest rate ($r > 0$).

  • $n$: Number of times interest is compounded each year.

  • $t$: Time in years.

• Compound Interest Example:

  • Investment: $\$1000$ at $7\%$ compounded annually ($n=1$) for $18$ years.

  • Calculation: $A = 1000(1 + 0.07)^{1 \cdot 18}$.

  • $A = 1000(1.07)^{18} \approx \$3379.93$.

• Key Concept: Equation for Exponential Decay:

  • Equation: $y = a(1 - r)^t$.

  • $a$: Initial amount.

  • $y$: Final amount.

  • $r$: Rate of decay as a decimal ($0 < r < 1$).

  • $t$: Time.

• Exponential Decay Application: Charity Donations:

  • Initial donations: $\$390,000$.

  • Drop rate: $1.1\%$ per year ($r = 0.011$).

  • Equation: $y = 390,000(1 - 0.011)^t = 390,000(0.989)^t$.

  • Estimate after 5 years: $y = 390,000(0.989)^5 \approx \$369,017$.

Questions & Discussion

• Substitution/Elimination Question: Solve system $r - t = -5$, $r + t = 25$. (Answer: $(10, 15)$).

• Substitution/Elimination Question: Solve system $2x + y = 7$, $y = 0.5x + 2$. (Answer: $(2, 3)$).

• Word Problem: Tens digit of a two-digit number is 5 more than twice the ones digit. Sum of digits is 8. What is the number? (Calculation: Let $t = 2o + 5$. $t + o = 8 \rightarrow 2o + 5 + o = 8 \rightarrow 3o = 3 \rightarrow o = 1$. Number is 71).

• Division Checkpoint: Simplify $\frac{a^3b^9}{ab^2}$. Answer: $a^{3-1}b^{9-2} = a^2b^7$.

• Power Checkpoint: Simplify $[(4^2)^2]^3$. Answer: $4^{2 \cdot 2 \cdot 3} = 4^{12}$.

• Scientific Notation Checkpoint: Express $0.0000452$ in scientific notation. Answer: $4.52 \times 10^{-5}$.