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Big principles Math book Algebra 1 Answer crucial Chapter 6 Exponential Functions and Sequences
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Exponential Functions and also Sequences keeping Mathematical Proficiency
Evaluate the expression.
Question 1.12(\(\frac142\)) – 33 + 15 – 92
Question 2.53 • 8 ÷ 22 + 20 • 3 – 4
Question 3.-7 + 16 ÷ 24 + (10 – 42)
Find the square root(s).
Question 8.12, 14, 16, 18, . . .
Question 9.6, 3, 0, -3, . . .
Question 10.22, 15, 8, 1, . . .
Question 11.ABSTRACT REASONINGRecall the a perfect square is a number with integers as its square roots. Is the product of 2 perfect squares always a perfect square? Is the quotient of two perfect squares constantly a perfect square? define your reasoning.
Exponential Functions and Sequences mathematical Practices
Question 1.A rabbit populace over 8 consecutive year is provided by 50, 80, 128, 205, 328, 524, 839, 1342. Uncover the populace in the tenth year.
Question 2.The sums of the numbers in the first eight rows of Pascal’s Triangle room 1, 2, 4, 8, 16, 32, 64, 128. Discover the sum of the numbers in the tenth row.
Lesson 6.1 properties of Exponents
Essential QuestionHow have the right to you write basic rules entailing properties that exponents?
EXPLORATION 1Writing Rules for Properties that ExponentsWork through a partner.a. What happens once you multiply two powers v the exact same base? create the product of the two powers as a single power. Then compose a general rule for finding the product of two powers with the very same base.
Communicate your Answer
Question 2.How can you write general rules entailing properties the exponents?
Question 3.There room 33 small cubes in the cube below. Create an expression for the variety of small cubes in the big cube in ~ the right.
Evaluate the expression.
Question 4.Simplify the expression
Simplify the expression. Compose your price using just positive exponents.
Question 5.104 • 10-6
Question 6.x9 • x-9
Simplify the expression. Write your prize using just positive exponents.
Question 13.(\(\frac12 k^2\))5
Question 14.(\(\frac6 c7\))-2
Question 15.Write two expressions that represent the area the a basic of the cylinder in instance 5.
Question 16.It bring away the Sun about 2.3 × 108 years to orbit the facility of the Milky Way. That takes Pluto about 2.5 × 102 years to orbit the Sun. How countless times does Pluto orbit the sunlight while the sun completes one orbit about the facility of the Milky Way? create your answer in clinical notation.
Properties of index number 6.1 Exercises
Vocabulary and also Core principle Check
Question 1.VOCABULARYWhich definitions or properties would certainly you use to simplify the expression (48 • 4-4)-2? Explain.Answer:
Question 2.WRITINGExplain when and also how to use the power of a Product Property.Answer:
Question 3.WRITINGExplain when and how to usage the Quotient of strength Property.Answer:
Question 4.DIFFERENT WORDS, exact same QUESTIONWhich is different? discover “both” answers.
In practice 5–12, advice the expression. (See instance 1.)
In practice 13–22, leveling the expression. Compose your answer using just positive exponents.
Question 17.\(\frac2^-2 m^-3n^0\)Answer:
Question 18.\(\frac10^0 r^-11 s3^2\)Answer:
Question 19.\(\frac4^-3 a^0b^-7\)Answer:
Question 20.\(\fracp^-87^-2 q^-9\)Answer:
Question 21.\(\frac2^2 y^-68^-1 z^0 x^-7\)Answer:
Question 22.\(\frac13 x^-5 y^05^-3 z^-10\)Answer:
In practice 23–32, leveling the expression. Create your price using just positive exponents.
Question 25.(-9)2 • (-9)2Answer:
Question 26.4-5 • 45Answer:
Question 29.6-8 • 65Answer:
Question 30.-7 • (-7)-4
Question 31.\(\fracx^5x^4\) • xAnswer:
Question 32.\(\fracz^8 \cdot z^2z^5\)Answer:
Question 33.USING PROPERTIESA microscopic lense magnifies an item 105 times. The size of things is 10-7 meter. What is its intensified length?
Question 34.USING PROPERTIESThe area of the rectangular computer system chip is 1123b2 square microns. What is the length?
ERROR ANALYSISIn practice 35 and 36, describe and correct the error in simplifying the expression.
In practice 37–44, simplify the expression. Create your prize using just positive exponents.
Question 44.(\(\frac12 r^6\))-6
Question 45.USING PROPERTIESWhich that the expressions represent the volume of the sphere? Explain.
Question 46.MODELING with MATHEMATICSDiffusion is the movement of molecules from one ar to another. The moment t (in seconds) that takes molecules to diffuse a distance of x centimeters is provided by t = \(\fracx^22 D\), where D is the diffusion coefficient. The diffusion coefficient because that a autumn of ink in water is around 10-5 square centimeters per second. Exactly how long will certainly it take the ink to diffuse 1 micrometer (10-4 centimeter)?Answer:
In exercises 47–50, leveling the expression. Write your answer using just positive exponents.
In practice 51–54, evaluate the expression. Write your price in clinical notation and also standard form.
Question 51.(3 × 102)(1.5 × 10-5)Answer:
Question 52.(6.1 × 10-3)(8 × 109)Answer:
Question 55.PROBLEM SOLVINGIn 2012, on average, around 9.46 × 10-1 pound of potatoes was developed for every 2.3 × 10-5 acre harvested. How countless pounds of potatoes on typical were created for each acre harvested? create your price in scientific notation and in conventional form.Answer:
Question 56.PROBLEM SOLVINGThe rate of light is about 3 × 105 kilometers per second. Just how long does it take sunshine to reach Jupiter? write your prize in clinical notation and in traditional form.
Question 57.MATHEMATICAL CONNECTIONSConsider Cube A and Cube B.
Question 58.PROBLEM SOLVINGA byte is a unit provided to measure up a computer’s memory. The table mirrors the number of bytes in numerous units that measure.
REWRITING EXPRESSIONSIn practice 59–62, rewrite the expression together a power of a product.
Question 63.USING STRUCTUREThe probability of rolling a 6 on a number cube is \(\frac16\). The probability of rolling a 6 double in a row is (\(\frac16\))2 = \(\frac136\).a. Write an expression that represents the probability of rolling a 6 n time in a row.
Question 64.HOW carry out YOU view IT?The shaded component of number n represents the part of a item of document visible ~ folding the file in fifty percent n times.
Question 65.REASONINGFind x and y as soon as \(\fracb^xb^y\) = b9 and also \(\fracb^x \cdot b^2b^3 y\) = b13. Define how you uncovered your answer.Answer:
Question 66.THOUGHT PROVOKINGWrite expressions for r and also h so the the volume the the cone have the right to be represented by the expression 27πx8. Uncover r and also h.
Question 67.MAKING an ARGUMENTOne that the smallest plant seeds comes from an orchid, and also one the the biggest plant seeds originates from a double coconut palm. A seed from an orchid has actually a fixed of 10-6 gram. The massive of a seed native a twin coconut palm is 1010 times the mass of the seed from the orchid. Her friend claims that the seed native the double coconut palm has actually a massive of around 1 kilogram. Is her friend correct? Explain.Answer:
Question 68.CRITICAL THINKINGYour school is conducting a survey. Students can answer the concerns in either part with “agree” or “disagree.”
Question 69.ABSTRACT REASONINGCompare the values of an and a-n once n 0 for(a) a > 1 and(b) 0 Answer:
Maintaining mathematical Proficiency
Find the square root(s).
Question 72.± \(\sqrt\frac164\)Answer:
Classify the genuine number in as numerous ways together possible.
Lesson 6.2 Radicals and Rational Exponents
Essential QuestionHow can you write and evaluate an nth source of a number?Recall the you cube a number as follows.
EXPLORATION 1Finding Cube RootsWork through a partner. use a cube source symbol to compose the side length of each cube. Then discover the cube root. Examine your answers by multiplying. I beg your pardon cube is the largest? Which 2 cubes are the same size? explain your reasoning.
EXPLORATION 2Estimating nth RootsWork through a partner. calculation each optimistic nth root. Then enhance each nth root v the point on the number line. Justify her answers.
Communicate your Answer
Question 3.How can you write and also evaluate an nth source of a number?
Question 4.The body mass m (in kilograms) the a dinosaur that walked on 2 feet can be modeled bym = (0.00016)C2.73where C is the one (in millimeters) that the dinosaur’s femur. The massive of a Tyrannosaurus rex was 4000 kilograms. Use a calculator to almost right the one of its femur.
Find the shown real nth root(s) the a.
Question 1.n = 3, a = -125
Question 2.n = 6, a = 64
Evaluate the expression.
Question 4.(-64)2 / 3
Question 5.95 / 2
Question 6.2563 / 4
Question 7.WHAT IF? In instance 4, the volume the the beach round is 17,000 cubic inches. Discover the radius come the nearest inch. Use 3.14 for π.
Question 8.The average cost of college tuition increases from $8500 to $13,500 over a duration of 8 years. Discover the yearly inflation rate to the nearest tenth of a percent.
Radicals and Rational index number 6.2 Exercises
Monitoring Progress and Modeling with Mathematics
Question 1.WRITINGExplain just how to advice 811 / 4.Answer:
Question 2.WHICH ONE doesn’t BELONG?Which expression does no belong with the various other three? describe your reasoning.
Monitoring Progress and also Modeling through Mathematics
In practice 3 and 4, rewrite the expression in rational exponent form.
In exercises 5 and also 6, rewrite the expression in radical form.
Question 5.151 / 3Answer:
Question 6.1401 / 8Answer:
In practice 7–10, find the indicated real nth root(s) that a.
Question 7.n = 2, a = 36Answer:
Question 8.n = 4, a = 81Answer:
Question 9.n = 3, a = 1000Answer:
Question 10.n = 9, a = -512Answer:
MATHEMATICAL CONNECTIONSIn practice 11 and 12, find the size of the cube. Check your answer.
In exercises 13–18, advice the expression.
Question 17.1281 / 7Answer:
Question 18.(-64)1 / 2Answer:
In exercises 19 and 20, rewrite the expression in reasonable exponent form.
In practice 21 and also 22, rewrite the expression in radical form.
Question 21.(-4)2 / 7Answer:
Question 22.95 / 2Answer:
In practice 23–28, evaluate the expression.
Question 23.323 / 5Answer:
Question 24.1252 / 3Answer:
Question 25.(-36)3 / 2Answer:
Question 26.(-243)2 / 5Answer:
Question 27.(-128)5 / 7Answer:
Question 28.3434 / 3Answer:
Question 29.ERROR ANALYSISDescribe and also correct the error in rewriting the expression in reasonable exponent form.
Question 30.ERROR ANALYSIS Describe and also correct the error in assessing the expression.
In exercises 31–34, advice the expression.
Question 31.(\(\frac11000\))1 / 3Answer:
Question 32.(\(\frac164\))1 / 6Answer:
Question 33.(27)-2 / 3Answer:
Question 34.(9)-5 / 2Answer:
Question 35.PROBLEM SOLVINGA math club is having a bake sale. Discover the area that the roasted sale sign.
Question 36.PROBLEM SOLVINGThe volume of a cube-shaped crate is 275 cubic millimeters. Uncover the length of one side of the box.Answer:
Question 37.MODELING through MATHEMATICS The radius r that the basic of a cone is provided by the equation r = (\(\frac3 V\pi h\))1 / 2
Question 38.MODELING v MATHEMATICSThe volume of a round is provided by the equation V = \(\frac16 \sqrt\pi\)S3 / 2, whereby S is the surface area that the sphere. Find the volume of a sphere, come the nearest cubic meter, that has a surface ar area the 60 square meters. Usage 3.14 for π.Answer:
Question 39.WRITINGExplain just how to create (\(\sqrt
Question 40.HOW perform YOU watch IT?Write an expression in rational exponent kind that to represent the side length of the square.
In practice 41 and 42, usage the formula r = (\(\fracFP\))1 / n – 1 to find the annual inflation price to the nearest tenth that a percent.
Question 41.A farm increases in worth from $800,000 come $1,100,000 over a duration of 6 years.Answer:
Question 42.The expense of a gallon that gas increases from $1.46 come $3.53 end a duration of 10 years.Answer:
Question 43.REASONINGFor what worths of x is x = x1 / 5?Answer:
Question 44.MAKING one ARGUMENTYour friend states that because that a actual number a and a positive integer n, the worth of \(\sqrt
In practice 45–48, simplify the expression.
Question 45.(y1 / 6)3 • xAnswer:
Question 46.(y • y1 / 3)3 / 2Answer:
Question 47.x • \(\sqrt<3>y^6\) + y2 • \(\sqrt<3>x^3\)Answer:
Question 48.(x1 / 3 • y1 / 2)9 • \(\sqrty\)Answer:
Question 49.PROBLEM SOLVINGThe formula for the volume that a continuous dodecahedron is V ≈ 7.66ℓ3, where ℓ is the length of an edge. The volume the the dodecahedron is 20 cubic feet. Calculation the sheet length.
Question 50.THOUGHT PROVOKINGFind a formula (for instance, indigenous geometry or physics) that consists of a radical. Rewrite the formula making use of rational exponents.Answer:
ABSTRACT REASONINGIn practice 51–56, permit x be a non an unfavorable real number. Determine whether the statement is always, sometimes, or never true. Justify her answer.Answer:
Question 51.(x1 / 3)3 = xAnswer:
Question 52.x1 / 3 = x-3Answer:
Question 53.x1 / 3 = \(\sqrt<3>x\)Answer:
Question 54.x = x1 / 3 • x3Answer:
Maintaining math Proficiency
Evaluate the role when x = −3, 0, and also 8.(Section 3.3)
Question 57.f(x) = 2x – 10Answer:
Question 58.w(x) = -5x – 1Answer:
Question 59.h(x) = 13 – xAnswer:
Question 60.g(x) = 8x + 16Answer:
Lesson 6.3 Exponential Functions
Essential QuestionWhat are some of the attributes of the graph of an exponential function?EXPLORATION 1Exploring one Exponential FunctionWork through a partner. Copy and complete each table because that the exponential function y = 16(2)x. In every table, what perform you an alert about the worths of x? What carry out you notice about the values of y?
EXPLORATION 2Exploring an Exponential FunctionWork with a partner. Repeat exploration 1 because that the exponential role y = 16(\(\frac12\))x. Carry out you think the statement below is true for any type of exponential function? Justify her answer.
EXPLORATION 3Graphing Exponential FunctionsWork through a partner. Sketch the graphs of the functions given in Explorations 1 and also 2. Just how are the graphs similar? how are lock different?
Communicate her Answer
Question 4.What are some of the characteristics of the graph of one exponential function?
Question 5.Sketch the graph of every exponential function. Does each graph have the characteristics you explained in question 4? explain your reasoning.a. Y = 2xb. Y = 2(3)xc. Y = 3(1.5)xd. Y = (\(\frac12\))xe. Y = 3(\(\frac12\))xf. Y = 2(\(\frac34\))x
Does the table stand for a straight or an exponential function? Explain.
Evaluate the function when x = −2, 0, and also \(\frac12\).
Question 3.y = 2(9)x
Question 4.y = 1.5(2)x
Graph the function. Compare the graph to the graph that the parental function. Describe the domain and range of f.
Question 5.f(x) = -2(4)x
Question 6.f(x) = 2(\(\frac14\))x
Graph the function. Explain the domain and range.
Question 7.y = -2(3)x + 2 – 1
Question 8.f(x) = (0.25)x + 3
Question 9.WHAT IF? In instance 6, the dependent change of g is multiplied by 3 for every 1 unit the independent variable x increases. Graph g once g(0) = 4. To compare g and the function f from example 3 over the term x = 0 come x = 2.
Question 10.A bacterial population y ~ x days deserve to be stood for by one exponential duty whose graph passes through (0, 100) and also (1, 200).(a) write a role that to represent the population.(b) uncover the population after 6 days.(c) does this bacterial population grow quicker than the bacterial populace in example 7? Explain.
Exponential functions 6.3 Exercises
Vocabulary and also Core principle Check
Question 1.OPEN-ENDEDSketch an increasing exponential function whose graph has actually a y-intercept of 2.Answer:
Question 2.REASONINGWhy is a the y-intercept that the graph the the function y = abx?Answer:
Question 3.WRITINGCompare the graph that y = 2(5)x v the graph the y = 5x.Answer:
Question 4.WHICH ONE no BELONG?Which equation does no belong v the various other three? explain your reasoning.
Monitoring Progress and Modeling v Mathematics
In exercises 5–10, recognize whether the equation represents an exponential function. Explain.
Question 5.y = 4(7)xAnswer:
Question 6.y = -6xAnswer:
Question 7.y = 2x3Answer:
Question 8.y = -3xAnswer:
Question 9.y = 9(-5)xAnswer:
Question 10.y = \(\frac12\)(1)xAnswer:
In exercises 11–14, identify whether the table represents a straight or an exponential function. Explain.
In practice 15–20, evaluate the function for the offered value that x.
Question 15.y = 3x; x = 2Answer:
Question 16.f(x) = 3(2)x; x = -1Answer:
Question 17.y = -4(5)x; x = 2Answer:
Question 18.f(x) = 0.5x; x = -3Answer:
Question 19.f(x) = \(\frac13\)(6)x; x = 3Answer:
Question 20.y = \(\frac14\)(4)x; x = \(\frac32\)Answer:
USING STRUCTUREIn practice 21–24, complement the function with its graph.
Question 21.f(x) = 2(0.5)xAnswer:
Question 22.y = -2(0.5)xAnswer:
Question 23.y = 2(2)xAnswer:
Question 24.f(x) = -2(2)x
In exercises 25–30, graph the function. To compare the graph come the graph that the parental function. Explain the domain and variety of f.
Question 25.f (x) = 3(0.5)xAnswer:
Question 26.f(x) = -4xAnswer:
Question 27.f(x) = -2(7)xAnswer:
Question 28.f(x) = 6 (\(\frac13\))xAnswer:
Question 29.f(x) = \(\frac12\)(8)xAnswer:
Question 30.f (x) = \(\frac32\)(0.25)xAnswer:
In practice 31–36, graph the function. Describe the domain and also range.
Question 31.f(x) = 3x – 1Answer:
Question 32.f(x) = 4x + 3Answer:
Question 33.y = 5x – 2 + 7Answer:
Question 34.y = – (\(\frac12\))x + 1 – 3Answer:
Question 35.y = -8(0.75)x + 2 – 2Answer:
Question 36.f(x) = 3(6)x – 1Answer:
In practice 37–40, compare the graphs. Discover the value of h, k, or a.
Question 41.ERROR ANALYSISDescribe and correct the error in analyzing the function.
Question 42.ERROR ANALYSISDescribe and correct the error in finding the domain and range of the function.
In practice 43 and also 44, graph the function with the provided description. Compare the function to f (x) = 0.5(4)x end the expression x = 0 come x = 2.
Question 43.An exponential role g models a connection in i beg your pardon the dependent variable is multiply by 2.5 because that every 1 unit the independent change x increases. The value of the function at 0 is 8.Answer:
Question 44.An exponential function h models a connection in i m sorry the dependent change is multiplied by \(\frac12\) because that every 1 unit the independent change x increases. The worth of the role at 0 is 32.Answer:
Question 45.MODELING with MATHEMATICSYou graph one exponential function on a calculator. Girlfriend zoom in continuously to 25% of the screen size. The role y = 0.25x represents the percent (in decimal form) of the original screen display screen that you see, where x is the number of times you zoom in.a. Graph the function. Define the domain and also range.b. Find and interpret the y-intercept.c. Friend zoom in twice. What percent that the original screen do you see?Answer:
Question 46.MODELING through MATHEMATICSA population y that coyotes in a national park triples every 20 years. The function y = 15(3)x to represent the population, wherein x is the variety of 20-year periods.
In exercises 47–50, write an exponential function represented by the table or graph.
Question 51.MODELING with MATHEMATICSThe graph to represent the number y of travellers to a brand-new art collection after x months.
Question 52.PROBLEM SOLVINGA sales report mirrors that 3300 gas grills were purchased indigenous a chain the hardware stores last year. The save expects grill sales to rise 6% every year. Around how many grills walk the keep expect to market in Year 6? use an equation to justify her answer.Answer:
Question 53.WRITINGGraph the duty f(x) = -2x. Climate graph g(x) = -2x – 3. How are the y-intercept, domain, and variety affected by the translation?Answer:
Question 54.MAKING one ARGUMENTYour friend states that the table to represent an exponential duty because y is multiplied by a constant factor. Is your friend correct? Explain.
Question 55.WRITINGDescribe the effect of a ~ above the graph of y = a • 2x once a is positive and also when a is negative.Answer:
Question 56.OPEN-ENDEDWrite a function whose graph is a horizontal translate into of the graph that h(x) = 4x.Answer:
Question 57.USING STRUCTUREThe graph the g is a translate into 4 systems up and also 3 units best of the graph of f(x) = 5x. Create an equation for g.Answer:
Question 58.HOW do YOU check out IT? The exponential role y = V(x) to represent the projected worth of a share x weeks after a copy, group loses an important legal battle. The graph the the role is shown.
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Question 59.USING GRAPHSThe graph to represent the exponential role f. Uncover f(7).