Road to 800

Exponents and exponential functions

Chapter 10 · 5 lessons · 224 questions

Exponents and radicals

Key points

  • Same base, multiplied: add the exponents. xa⋅xbx^a \cdot x^b =xa+b= x^{a + b}.
  • Same base, divided: subtract the exponents. A power of a power: multiply the exponents.
  • A root is a fractional exponent: xmn=xmn\sqrt[n]{x^m} = x^{\dfrac{m}{n}}. The index of the root goes in the denominator.
  • A negative exponent means ‘one over’: x−a=1xax^{-a} = \dfrac{1}{x^a}.
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See it solved

Example

Which expression is equivalent to x53⋅x13\sqrt[3]{x^5} \cdot x^{\dfrac{1}{3}}, where x>0x > 0?

  1. Ax59x^{\dfrac{5}{9}}
  2. Bx1415x^{\dfrac{14}{15}}
  3. Cx43x^{\dfrac{4}{3}}
  4. Dx2x^2

Worked by hand

The plan

  1. Turn the root into a fractional exponent
  2. Use the exponent rules

The exponent rules are the method. Desmos is a check: put a number in place of xx and compare the values.

  1. Step 1: Turn the root into a fractional exponent

    • x53=x53\sqrt[3]{x^5} = x^{\dfrac{5}{3}}. The index 33 goes in the denominator.

  2. Step 2: Use the exponent rules

    • The expression is x53⋅x13x^{\dfrac{5}{3}} \cdot x^{\dfrac{1}{3}}.
    • Same base, multiplied: add the exponents.
    • 53+13=63=2\dfrac{5}{3} + \dfrac{1}{3} = \dfrac{6}{3} = 2.
    • The result is x2x^2.
The correct answer choice is D: x2x^2.

The same example in Desmos

  1. new graph

    Keys: 8^5/3 → → *8^1/3

    6464: the given expression with 88 in place of xx.

  2. 6464 too. Choice D matches.

Type
Desmos expression list: 8^{frac{5}{3}}·8^{frac{1}{3}}; 8^2

Made with Desmos

Open Desmos

Your turn

2x+12^{x + 1} =8x−1= 8^{x - 1}

What is the solution to the given equation?

Show the answerHide the answer

Answer

The correct answer is 22.

  1. Step 1: Rewrite both sides on a common base

    • 8=238 = 2^3, so 8x−18^{x - 1} =(23)x−1= (2^3)^{x - 1}.
    • A power of a power: multiply the exponents. 3(x−1)3(x - 1) =3x−3= 3x - 3.
    • The equation is 2x+12^{x + 1} =23x−3= 2^{3x - 3}.

  2. Step 2: Equate the exponents

    • x+1=3x−3x + 1 = 3x - 3.
    • Subtract xx from both sides: 1=2x−31 = 2x - 3.
    • Add 33 to both sides: 4=2x4 = 2x, so x=2x = 2.
    • Check: 22+12^{2 + 1} =23= 2^3 =8= 8 and 82−18^{2 - 1} =81= 8^1 =8= 8. It matches.

Traps

  • Multiplying the exponents when the bases are multiplied. 53⋅13=59\dfrac{5}{3} \cdot \dfrac{1}{3} = \dfrac{5}{9} is the rule for a power of a power. That is choice A, a wrong answer.
  • Flipping the fractional exponent. x53\sqrt[3]{x^5} is x53x^{\dfrac{5}{3}}, not x35x^{\dfrac{3}{5}}. The flipped version gives 35+13=1415\dfrac{3}{5} + \dfrac{1}{3} = \dfrac{14}{15}: choice B, a wrong answer.
Practice this lesson56 questions

Linear or exponential?

Key points

  • Linear: the amount changes by the same number each step. You add or subtract.
  • Exponential: the amount changes by the same factor or percent each step. You multiply.
  • Cue words for exponential: doubles, halves, triples, a percent of the previous amount.
  • In a table with equal steps in xx, equal differences mean linear and equal ratios mean exponential.
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See it solved

Example

The table shows four values of xx and their corresponding values of yy. Which statement describes the relationship between xx and yy?

xx0123
yy51545135
  1. AIt is linear: yy increases by 1010 each time xx increases by 11.
  2. BIt is linear: yy increases by 3030 each time xx increases by 11.
  3. CIt is exponential: yy is multiplied by 33 each time xx increases by 11.
  4. DIt is exponential: yy is multiplied by 1010 each time xx increases by 11.

Worked by hand

The plan

  1. Check the differences
  2. Check the ratios

Differences and ratios take a few seconds by hand. Desmos is a check: an exponential fit gives the factor.

  1. Step 1: Check the differences

    • 15−5=1015 - 5 = 10.
    • 45−15=3045 - 15 = 30.
    • 135−45=90135 - 45 = 90.
    • The differences are not equal, so the relationship is not linear. That rules out choices A and B.

  2. Step 2: Check the ratios

    • 155=3\dfrac{15}{5} = 3.
    • 4515=3\dfrac{45}{15} = 3.
    • 13545=3\dfrac{135}{45} = 3.
    • The ratios are equal, so the relationship is exponential, and the factor is 33: choice C.
The correct answer choice is C: the factor is 33, so each value of yy is 33 times the one before.

The same example in Desmos

  1. new graph, then Add item + and Table

    Table
    x1y1
    05
    115
    245
    3135

    Four points. Zoom out to see them all.

  2. Keys: y1~ab^x1

    a=5a = 5 and b=3b = 3: start at 55 and multiply by 33 each step.

Type
Desmos expression list: a table of x1: 0, 1, 2, 3 and y1: 5, 15, 45, 135; y1~ ab^{x1
Look
Desmos graph of a table of x1: 0, 1, 2, 3 and y1: 5, 15, 45, 135; y1~ ab^{x1

Made with Desmos

Open Desmos

Your turn

A savings jar holds $200\$200. Each month, $25\$25 is added to the jar. Is the growth of the amount in the jar linear or exponential, and how much is in the jar after 66 months?

Show the answerHide the answer

Answer

The growth is linear, and the jar holds $350\$350 after 66 months.

  1. Step 1: Determine the relationship

    • The same amount, $25\$25, is added each month.
    • Adding the same amount each step is linear.

  2. Step 2: Determine the amount

    • 25⋅6=15025 \cdot 6 = 150 dollars are added in 66 months.
    • 200+150=350200 + 150 = 350 dollars.

Traps

  • Calling any fast growth exponential. Growth is exponential only when each step multiplies by the same factor. Here the differences are 1010, 3030 and 9090, but every ratio is 33.
  • Reading the first difference as the rule. 15−5=1015 - 5 = 10 suggests ‘add 1010’, which is choice A. The next step adds 3030, so that rule fails.
Practice this lesson12 questions

Exponential growth and decay

Key points

  • An exponential model is y=a(b)xy = a(b)^x: start at aa, then multiply by bb once each time interval.
  • For growth by a rate rr each interval, b=1+rb = 1 + r. For decay, b=1−rb = 1 - r. Write rr as a decimal.
  • b>1b > 1 is growth. bb between 00 and 11 is decay.
  • The exponent counts the intervals: the time divided by the length of one interval.
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See it solved

Example

A town has 8,0008{,}000 people. Its population grows by 5%5\% each year. Which function gives the population P(t)P(t) after tt years?

  1. AP(t)P(t) =8,000(0.05)t= 8{,}000(0.05)^t
  2. BP(t)P(t) =8,000(0.95)t= 8{,}000(0.95)^t
  3. CP(t)P(t) =8,000(1.05)t= 8{,}000(1.05)^t
  4. DP(t)P(t) =8,000(1.5)t= 8{,}000(1.5)^t

Worked by hand

The plan

  1. Determine the components of the exponential equation
  2. Check with one interval

Writing the model is done by hand. Desmos is a check: define the model and look at its value after one interval.

  1. Step 1: Determine the components of the exponential equation

    • The initial amount is a=8,000a = 8{,}000.
    • The rate is 5%5\%. As a decimal, r=0.05r = 0.05.
    • The population grows, so bb =1+r= 1 + r =1+0.05= 1 + 0.05 =1.05= 1.05.
    • That rules out choice A (the rate alone), choice B (decay) and choice D (50%50\% growth).
    • The exponent is tt, because the population is multiplied once each year.
    • P(t)P(t) =8,000(1.05)t= 8{,}000(1.05)^t.

  2. Step 2: Check with one interval

    • After 11 year, 5%5\% of 8,0008{,}000 is 0.05⋅8,0000.05 \cdot 8{,}000 =400= 400.
    • 8,000+4008{,}000 + 400 =8,400= 8{,}400.
    • The model gives 8,000⋅1.058{,}000 \cdot 1.05 =8,400= 8{,}400. It matches.
The correct answer choice is C: P(t)P(t) =8,000(1.05)t= 8{,}000(1.05)^t.

The same example in Desmos

  1. new graph

    Keys: f(x)=8000(1.05)^x

    The model for choice C, with xx for the years.

  2. 84008400, at the right of the row. After one year the population is 8,4008{,}400, which is 5%5\% more.

Type
Desmos expression list: f(x)=8000(1.05)^x; f(1)

Made with Desmos

Open Desmos

Your turn

A car is worth $20,000\$20{,}000. It loses 10%10\% of its value each year. What is the car worth after 22 years?

Show the answerHide the answer

Answer

The correct answer is $16,200\$16{,}200.

  1. Step 1: Determine the components of the exponential equation

    • The initial amount is a=20,000a = 20{,}000.
    • The rate is 10%10\%, so r=0.10r = 0.10. The value goes down, so bb =1−0.10= 1 - 0.10 =0.90= 0.90.
    • 22 years is 22 intervals.

  2. Step 2: Determine the final value

    • The value is 20,000⋅0.90220{,}000 \cdot 0.90^2.
    • 0.902=0.810.90^2 = 0.81.
    • 20,000⋅0.8120{,}000 \cdot 0.81 =16,200= 16{,}200. The car is worth $16,200\$16{,}200.

Traps

  • Using the rate as the base. (0.05)t(0.05)^t multiplies by 0.050.05 each year, which leaves only 5%5\% of the people. That is choice A, a wrong answer.
  • Subtracting the same amount each year. 10%10\% of 20,00020{,}000 is 2,0002{,}000, but the second year's loss is 10%10\% of 18,00018{,}000, which is 1,8001{,}800. 20,000−4,00020{,}000 - 4{,}000 =16,000= 16{,}000 is a wrong answer.
Practice this lesson88 questions

Rewriting an exponential: period and starting point

Key points

  • The exponent counts how many times the factor is applied.
  • If the factor is applied once every kk years, the exponent is tk\dfrac{t}{k}. Doubling every 33 years is 2t32^{\dfrac{t}{3}}.
  • To change years to months, write tt years as m12\dfrac{m}{12}, where mm is the number of months.
  • The percent rate comes from the base: 1.041.04 is 4%4\% growth, and 0.850.85 is 15%15\% decay.
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See it solved

Example

A dish holds 500500 bacteria. The number of bacteria doubles every 33 hours. Which function gives the number of bacteria N(h)N(h) after hh hours?

  1. AN(h)=500(2)3hN(h) = 500(2)^{3h}
  2. BN(h)=500(2)h3N(h) = 500(2)^{\dfrac{h}{3}}
  3. CN(h)=500(3)h2N(h) = 500(3)^{\dfrac{h}{2}}
  4. DN(h)=500(2)hN(h) = 500(2)^{h}

Worked by hand

The plan

  1. Determine the components of the exponential equation
  2. Determine the number of intervals

Decide the exponent by hand. Desmos is a check: define a choice and see whether its values fit the story.

  1. Step 1: Determine the components of the exponential equation

    • The initial amount is 500500.
    • The number doubles, so the factor is 22.
    • That rules out choice C, which has a factor of 33.

  2. Step 2: Determine the number of intervals

    • One interval is 33 hours. In hh hours there are h3\dfrac{h}{3} intervals.
    • So the exponent is h3\dfrac{h}{3}, and N(h)=500(2)h3N(h) = 500(2)^{\dfrac{h}{3}}.
    • That rules out choices A and D.
    • Check: at h=3h = 3 the exponent is 33=1\dfrac{3}{3} = 1, and 500⋅2=1,000500 \cdot 2 = 1{,}000. The number has doubled once.
The correct answer choice is B: N(h)=500(2)h3N(h) = 500(2)^{\dfrac{h}{3}}.

The same example in Desmos

  1. new graph

    Keys: f(x)=500(2)^x/3

    The model for choice B, with xx for the hours.

  2. 10001000: after 33 hours the number has doubled once.

  3. 20002000: after 66 hours it has doubled twice.

Type
Desmos expression list: f(x)=500(2)^{frac{x}{3}}; f(3); f(6)

Made with Desmos

Open Desmos

Your turn

The value of a machine, in dollars, tt years after it is bought is V(t)V(t) =1,200(0.85)t= 1{,}200(0.85)^t. By what percent does the value decrease each year?

Show the answerHide the answer

Answer

The correct answer is 15%15\%.

  1. Step 1: Determine the components of the exponential equation

    • The base is b=0.85b = 0.85. It is less than 11, so the value decays.
    • For decay, b=1−rb = 1 - r. So rr =1−0.85= 1 - 0.85 =0.15= 0.15.
    • As a percent, 0.150.15 is 15%15\%.

Traps

  • Multiplying the exponent instead of dividing. 23h2^{3h} doubles 33 times every hour. That is choice A, a wrong answer.
  • Ignoring the interval. 2h2^{h} doubles every hour, not every 33 hours. That is choice D, a wrong answer.
Practice this lesson40 questions

Graphs of exponential functions

Key points

  • The graph of y=a(b)xy = a(b)^x crosses the yy-axis at (0,a)(0, a), because b0=1b^0 = 1.
  • It also passes through (1,ab)(1, ab).
  • If b>1b > 1, the curve rises from left to right. If bb is between 00 and 11, it falls.
  • Adding a constant, y=a(b)x+cy = a(b)^x + c, moves the whole curve up cc units. The yy-intercept becomes (0,a+c)(0, a + c), and the curve flattens out toward y=cy = c.
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See it solved

Example

The function ff is defined by f(x)f(x) =3(2)x+4= 3(2)^x + 4. What is the yy-coordinate of the yy-intercept of the graph of y=f(x)y = f(x) in the xyxy-plane?

Worked by hand

The plan

  1. Determine the components of the exponential equation
  2. Plug the input into the function

Desmos is quicker here: type the equation and click the point where the curve meets the yy-axis.

  1. Step 1: Determine the components of the exponential equation

    • a=3a = 3 and b=2b = 2. The constant added is 44.

  2. Step 2: Plug the input into the function

    • The yy-intercept is at x=0x = 0.
    • f(0)f(0) =3(2)0+4= 3(2)^0 + 4.
    • 20=12^0 = 1, so 3⋅1=33 \cdot 1 = 3.
    • f(0)=3+4=7f(0) = 3 + 4 = 7.
The correct answer is 77.

The same example in Desmos

  1. new graph

    Keys: y=3(2)^x → +4

    A rising curve.

  2. click the point where the curve meets the y-axis

    (0,7)(0, 7). The yy-coordinate of the yy-intercept is 77.

Type
Desmos expression list: y=3(2)^x+4
Look
Desmos graph of y=3(2)^x+4, with (0, 7) labelled

Made with Desmos

Open Desmos

Your turn

In the xyxy-plane, the graph of y=a(b)xy = a(b)^x passes through the points (0,5)(0, 5) and (1,15)(1, 15), where aa and bb are constants. What is the value of bb?

Show the answerHide the answer

Answer

The correct answer is 33.

  1. Step 1: Determine the value of aa

    • At x=0x = 0: yy =a⋅b0= a \cdot b^0 =a⋅1= a \cdot 1 =a= a.
    • The graph passes through (0,5)(0, 5), so a=5a = 5.

  2. Step 2: Determine the value of bb

    • At x=1x = 1: y=a⋅b1=5by = a \cdot b^1 = 5b.
    • The graph passes through (1,15)(1, 15), so 5b=155b = 15.
    • b=155=3b = \dfrac{15}{5} = 3.

Traps

  • Forgetting the constant. 3(2)0=33(2)^0 = 3 is not the whole output. Add the 44: the yy-coordinate is 77, not 33.
  • Taking 202^0 as 00. Any nonzero number to the power 00 is 11. With 00, the answer would be 0+4=40 + 4 = 4, a wrong answer.
Practice this lesson28 questions
Typing tips for Desmos
  • In the keys lines, → means the right-arrow key.
  • Typing ^ or sqrt works the same way: press → to come back down from an exponent or out of a square root before typing the rest.
  • Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
  • Type y1 and Desmos shows y₁: a digit typed straight after a letter becomes a subscript. If you type the underscore yourself (y_1), press → after the digit: a ~ typed next lands inside the subscript (+, − and = leave it by themselves).
  • The ~ key is Shift + ` on most keyboards (the key left of 1). Use ~, not =: ~ asks Desmos to find the unknown letters; = only draws or defines.
  • A letter that already has a value, such as a slider m = 1 left over from earlier, is held fixed by a ~ fit, with no warning. Start a new graph, or delete old sliders, before each question.
  • Start a new graph for each example: a letter or function defined twice gives the error 'You've defined … in more than one place'.
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