1.1 exponents t
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The document discusses exponent rules including: 1) The multiply-add rule for exponents (AnAk = An+k) and divide-subtract rule (An/Ak = An-k). 2) The power-multiply rule for exponents ((An)k = Ank). 3) Special exponent rules including the 0-power rule (A0 = 1), negative power rule (A-k = 1/Ak), and fractional power rules. 4) How to write radicals with fractional exponents (√an = (√a)n = an/n) and how to simplify expressions using fractional exponents. 5) How to write decimal exponents as fractional exponents in order to simplify expressions using exponent rules
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The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It presents rules for multiplying, dividing, and raising exponents. Examples are provided to demonstrate applying the rules, such as using the power-multiply rule to evaluate (22*34)3. Special exponent rules are also covered, such as the 0-power rule where A0 equals 1 when A is not 0. The document provides examples of calculating fractional exponents by first extracting the root and then raising it to the numerator power.
1.1 exponents yz
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It introduces common exponent rules like the multiply-add rule, divide-subtract rule, and power-multiply rule. It also covers special exponents like fractional exponents, zero exponents, negative exponents, and nth roots. Examples are provided to demonstrate applying the various exponent rules.
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The document discusses several rules and properties related to exponents:
1) When multiplying like bases, you add the exponents.
2) When raising a power to a power, you multiply the exponents.
3) Any integer raised to a negative one power is the reciprocal of that integer.
4) Any number raised to the first power is itself, and any number raised to the zero power is one.
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1 exponents yz
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It presents rules for multiplying, dividing, and raising exponents. Examples are provided to demonstrate applying the rules, such as using the power-multiply rule to evaluate (22*34)3. Special exponent rules are also covered, such as the 0-power rule where A0 equals 1 when A is not 0. The document provides examples of calculating fractional exponents by first extracting the root and then raising it to the numerator power.
1.1 exponents yz
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It introduces common exponent rules like the multiply-add rule, divide-subtract rule, and power-multiply rule. It also covers special exponents like fractional exponents, zero exponents, negative exponents, and nth roots. Examples are provided to demonstrate applying the various exponent rules.
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The document discusses several rules and properties related to exponents:
1) When multiplying like bases, you add the exponents.
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The document discusses the derivatives of exponential functions. It calculates the derivative of f(x)=2^x and finds that the derivative of 2^x is proportional to 2^x with a constant of proportionality of 0.693. It also shows values of the derivative for different values of a in f(x)=a^x and determines that the graph of these values follows the function y=ln(a).
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The document introduces exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples of evaluating exponents like 43. It then introduces rules for exponents, including the multiply-add rule where ANAK = AN+K, and the divide-subtract rule where AN/AK = AN-K. It also covers fractional exponents by defining the 0-power rule where A0 = 1 and the negative power rule where A-K = 1/AK.
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The interval with the greatest increase in heat is found to be from 0-3.2 minutes, with a maximum increase of 218.0 degrees F per minute at 0 minutes.
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1.1 exponents
This document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. It then provides several exponent rules including:
- Multiply-Add Rule: AnAk = An+k
- Divide-Subtract Rule: An/Ak = An-k
- Power-Multiply Rule: (An)k = Ank
Special exponents are also discussed such as A0=1 if A≠0, A-k=1/Ak, and calculating fractional exponents by extracting the root first then raising it to the numerator power. Examples are provided to demonstrate applying these exponent rules and calculating fractional exponents.
0. exponents y
The document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. The base is A and the exponent is N. Some key exponent rules covered include: the multiply-add rule where AnAk = An+k; the divide-subtract rule where An/Ak = An-k; the power-multiply rule where (An)k = Ank. Special exponents rules include: A0 = 1 when A ≠ 0; A-k = 1/Ak; A1/n = nth root of A. Examples are provided to demonstrate applying these exponent rules.
1.1 exponents y
The document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. The base is A and the exponent is N. Some key exponent rules covered include: the multiply-add rule where AnAk = An+k; the divide-subtract rule where An/Ak = An-k; the power-multiply rule where (An)k = Ank. Special exponents rules include: A0 = 1 when A ≠ 0; A-k = 1/Ak; A1/n = nth root of A. Examples are provided to demonstrate applying these exponent rules.
1.7 power equations and calculator inputs
The document reviews exponential notation and rules of exponents. It defines exponential notation as the quantity A multiplied to itself N times written as AN. It then presents rules for exponents, including the multiply-add rule, divide-subtract rule, and power-multiply rule. It extends the definition of exponents to include negative and fractional exponents through rules like the 0-power rule and negative power rule. It provides examples of simplifying expressions using these rules.
4 5 fractional exponents-x
This document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples like 43 = 64. Rules for exponents are covered, including:
- Multiply-Add Rule: ANAK = AN+K
- Divide-Subtract Rule: AN/AK = AN-K
- Negative exponents represent reciprocals, so A-K = 1/AK
Fractional exponents are introduced, along with the 0-Power Rule that A0 = 1.
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The document discusses the basics of exponential notations and rules of exponents. It defines exponential expressions where the base A is multiplied by itself N times as AN. It presents rules for multiplication, division, and powers of exponents. It also extends the definitions and rules to fractional, negative, decimal and real number exponents by writing them as fractions and applying the fractional exponent rules. Examples are provided to demonstrate simplifying expressions using these rules.
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Powers (or exponents) indicate how many times a base number is multiplied by itself. Some key properties of powers include:
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2) Scientific notation is a way of writing numbers as a decimal between 1 and 10 multiplied by a power of 10. It allows for easy representation of very large or small numbers. Numbers are converted between scientific and standard notation by moving the decimal point and adjusting the exponent accordingly.
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The document introduces exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples of evaluating exponents like 43. It then introduces rules for exponents, including the multiply-add rule where ANAK = AN+K, and the divide-subtract rule where AN/AK = AN-K. It also covers fractional exponents by defining the 0-power rule where A0 = 1 and the negative power rule where A-K = 1/AK.
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The document summarizes the heat output of a hair straightener over time. It is described by three functions between 0-3.2 minutes, 3.2-8 minutes, and 8-12 minutes.
The total heat output over the full 12 minutes is found to be 3224.98 hundred degree units by calculating the integrals of each function.
The interval with the greatest increase in heat is found to be from 0-3.2 minutes, with a maximum increase of 218.0 degrees F per minute at 0 minutes.
The heat decrease at 9.2 minutes is calculated to be 145.24 degrees F per minute by substituting into the third function.
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Two wattmeters connected to a three-phase motor measured readings of 5 kW and -1 kW. The total power was calculated as the sum of the wattmeter readings, which was 4 kW. The power factor was calculated from the wattmeter readings using trigonometric functions and was found to be 0.36.
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This document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. It then provides several exponent rules including:
- Multiply-Add Rule: AnAk = An+k
- Divide-Subtract Rule: An/Ak = An-k
- Power-Multiply Rule: (An)k = Ank
Special exponents are also discussed such as A0=1 if A≠0, A-k=1/Ak, and calculating fractional exponents by extracting the root first then raising it to the numerator power. Examples are provided to demonstrate applying these exponent rules and calculating fractional exponents.
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The document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. The base is A and the exponent is N. Some key exponent rules covered include: the multiply-add rule where AnAk = An+k; the divide-subtract rule where An/Ak = An-k; the power-multiply rule where (An)k = Ank. Special exponents rules include: A0 = 1 when A ≠ 0; A-k = 1/Ak; A1/n = nth root of A. Examples are provided to demonstrate applying these exponent rules.
1.1 exponents y
The document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. The base is A and the exponent is N. Some key exponent rules covered include: the multiply-add rule where AnAk = An+k; the divide-subtract rule where An/Ak = An-k; the power-multiply rule where (An)k = Ank. Special exponents rules include: A0 = 1 when A ≠ 0; A-k = 1/Ak; A1/n = nth root of A. Examples are provided to demonstrate applying these exponent rules.
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This document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples like 43 = 64. Rules for exponents are covered, including:
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The document discusses exponential functions and properties of exponents. It provides examples of simplifying expressions using properties like product of powers, power of a power, quotient of powers, and rational exponents. It also covers evaluating exponential functions, graphing them, and identifying linear vs exponential patterns in tables and graphs. The goal is for students to gain a conceptual understanding of working with exponential expressions and functions.
1.3 solving equations t
1. The document discusses various methods for solving different types of equations including: polynomial equations using factoring, quadratic equations using the quadratic formula, rational equations by clearing denominators, radical equations by squaring both sides, and absolute value equations geometrically.
2. Specific examples are provided to demonstrate how to solve equations involving polynomials, rationals, radicals, powers, and absolute values. Steps include factoring, using the quadratic formula, clearing denominators, isolating and squaring radicals, and raising both sides to appropriate powers.
3. The document also discusses finding zeros of rational functions and domains by determining where the numerator or denominator is equal to zero.
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1.1 exponents t
- 1. Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule: An Ak Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk Exponents = An – k Exponent–Rules 0-power Rule: A0 = 1 (A=0) = 1 Ak 1 Ak A0 Ak ½ - Power Rule: A½ = A , the square root of A, because (A½)2 = A = (A)2, so A½ = A because 1 = = A1–1 = A0A1 A1 Negative Power Rule: A–k = because = A0–k = A–k 1/n - Power Rule: A1/n = A , the nth root of A. n (divide–subtract) (divide–subtract)
- 2. Special Exponents Example B. c. 641/3 = 64 = 4 3 b. 81/3 = 8 = 2 3 a. 641/2 = 64 = 8 a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27 Example C. Find the root first, then raise the root to the numerator–power. The fractional exponent A k n± (A ) kn ±1 is 1. take the nth root of A 2. then raise the root to ±k power c. 16 -3/4 = (161/4)-3 = (16)-3 = (2)-3 = 1/23 = 1/84 b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9 3 1/n - Power Rule: A1/n = A , the nth root of A. n
- 3. Fractional Powers x*(x1/3y3/2)2 x–1/2y2/3 = x*x2/3y3 x–1/2y2/3 = x–1/2y2/3 = x5/3y3 x5/3 – (–1/2) y3 – 2/3 = x13/6 y7/3 Example D. Simplify by combining the exponents. power–multiply rule 1/3*2 3/2*2 multiply–add rule 1 + 2/3 divide–subtract rule To write a radical in fractional exponent form: an = ( a )n → a k k k n Example E. Write the following expressions using fractional exponents then simplify if possible. a2 a = a2/3a1/4 = a11/12 = a11 3 4 12 c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ). a. 53 or (5 )3 = 53/2 b. 9a2 = (9a2)1/2 = 3a d. Multiply Use the exponent-rules to collect exponents.
- 4. Decimal Powers We write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents. Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27 b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8 4 c. 30.4 = 32/5 = (3)2 ≈ 1.55 (by calculator) 5
- 8. (Answers to odd problems) Exercise A. 1. 1/2 3. 1/8 5.1/27 7. 1/64 9.3/2 11.1/18 13.1/6 Exercise B. 1. 𝑥8 3. 1/𝑥2 5.𝑥7/𝑦2 7. 1/𝑥4 9. 𝑥3 11. 1 𝑥7 13. 𝑦8 36𝑥6 15. 2 𝑥+3 17. 𝑎2𝑥+3 Exercise C. 1. 8 3. (−8) 2 3 5. 8/27 7. 16 9. 1/8 Exponents Exercise D. 1. 𝑥3 3. 8𝑥4 6 𝑦17 5. 6 𝑦 𝑥3 7. 3𝑎11 2𝑏5 9. 4−5 11. 5 𝑥16 13. 6 (2𝑥 + 1)13 15. 6 sin(𝑥)17 Exercise E. 1.𝑞/𝑝 3. −3 5. 13 7. −4 9. 2 ∙ 3 2 11. 4 1 𝑥+5 Exercise F. 1. 𝑥 = 4, 𝑦 = 2, 𝑧 = 3 3. 𝑥 = 3, 𝑦 = −1, 𝑧 = 6