The Power of Exponents Quiz | Math

Quiz Answer Key and Fun Facts

1. Gregory opens up a savings deposit in the bank and deposits an initial $1000. The bank is rather eccentric and reports the total amount of earnings in exponential equations. When Gregory returns to the bank one day, the bank clerk gives him this equation: 2^(x+4) = 8^(3x) Multiply x to your original deposit; this number is the amount of interest earned. Now add the amount of interest earned to the initial deposit to find the total amount of money in his savings deposit. How much money is in Gregory's saving deposit?

Answer: $1500

A way to solve this exponential equation is by getting all of the bases to be the same. The number "2" is can be a common base if you change the 8 to a 2^3.

2^(x+4) = 2^3^(3x)

You can multiply the exponents on the right side of the equation:

2^(x+4) = 2^(9x)

Since the bases on both side of the equation are now the same, you can just set the exponents equal to each other:

x+4 = 9x

With a bit of simple algebra, you will find that x = .5
If you multiply the initial deposit of $1000 by .5 you have the interest earned, which is $500. Adding that to the initial amount will give you Gregory's total amount of $1500.

2. Four students are at the last question in a math contest. Whoever answers this question correctly wins first place: If x>0, solve: x^x^x^x^x^... = 2 Basically the x is infinitely raised to the x power; it's an infinite power tower. Each of the students submit an answer: Samantha: e^pi Fredrick: e Ariel: 2^(1/2) Timothy: 3^(1/2) Assuming that each of the answers are different numerical values, who won first place in the contest?

Answer: Ariel

This is quite a tricky question that requires the use of logic more so than actual calculations. For the infinite power tower, if you remove the bottom x, it wouldn't change the equation at all because the x's in the power tower continue forever; it would still be the same. If you remove the bottom x the equation would be:

x^x^x^x^... = 2

It's still the same. Therefore, you can say that the exponents of the infinite power tower also equal 2. If you replace the exponents with two, the equation would become:

x^2 = 2

If you take the square root of both sides you can find in exponential form that x = 2^(1/2)

Ariel is the student that had the same answer, so she wins first place.

3. Jennifer goes to her local deli. She takes a ticket that has her place number. The number determines what order the customers will be served. For some strange reason her ticket says: Simply x. The exponent of x is your place: ((x^-1)*(x^3))/((x^6)*(x^-10)) What place is Jennifer?

Answer: 6

The x's on the bottom of the fraction can be moved to the top of the fraction by making their exponent negative. There's a property that explains this:

1/(a^b) = a^-b

So, the fraction then becomes:

(x^-1)*(x^3)*(x^-6)*(x^10)

Since all the bases are the same and are being multiplied together, you can just add the exponents up to simplify it.

x^(-1+3-6+10)

Which equals x^6. The exponent is 6, so Jennifer's place also is 6.

4. William is sending out applications for colleges. He's been thinking of applying to the School of Powers; however, the school only accepts students with HPA's (High School Point Average) greater than or equal to 5.5. William's HPA is the sum of coefficients in the expression below divided by 16. (a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 + h^2)^2 Does William qualify for the School of Powers?

Answer: No

In order to get the coefficients of the different variables in the expression, just set all the variables to 1.

(1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2)^2

This simply equals 8^2, which is 64. That number divided by 16 is 4. So, William's HPA is 4. Therefore, he does not qualify for the college.

5. Spencer is taking an exponents test in her math class. One of the questions asks her to write down all the debated values of 0^0. So, she writes down: 1 -1 undefined For each correct value she gets 3 points. How many points does she earn on this question?

Answer: 6

Generally, 0^0 is thought of as either 1 or undefined depending on which arguments you take. For 0^0 = 1, mathematicians argue that the binomial theorem would not be valid unless 0^0 = 1. Mathematicians who believe that the expression is undefined rely on limits to prove that 0^0 is undefined.

6. Brendon asks Alex to play a little game of chance. Brendon tells him that he must role a fair six-sided die. Whatever number Alex gets, he must plug it into: [x^9 + 7x^7 - x^6 + 3x^4 + 24x]^0 If the value Alex gets is higher than or equal to 10, he wins one dollar. But, if the value Alex gets is lower than 10, Brendon wins one dollar. Should Alex play this game?

Answer: No

If you carefully look at the expression, you should notice that everything inside the brackets is raised to the 0 power. And, anything to the 0 power is simply 1. So, no matter what number Alex rolls, the value he'll get will be 1. Therefore, he will lose every game he plays, so he should not play.

7. Alice has just lost a number of baby teeth and is planning to put them under her pillow so that the "Tooth Fairy" will reward her with spare change. Although she may believe in the Tooth Fairy, she's quite skilled at mathematics. Her mother wants to know how many teeth she's lost so that she can remember to keep that amount of quarters during the day. Alice writes the following down on a scrap piece of paper: 27^(3x) = 3^(5x+8) Find the value of x and you'll know how many teeth I've lost. How much money will Alice's mother need if she will give 25 cents for each tooth?

Answer: 50 cents

This problem is similar to the first one; it can be solved the same way. The equation must be manipulated so that the bases are the same. An easy base to change to would be 3.

3^(3^3x) = 3^(5x+8)

Multiply the two exponents to get:

3^(9x) = 3^(5x+8)

And then set the exponents equal to each other:

9x = 5x+8

Lastly, with simple algebra you can find that Alice has lost 2 teeth, so the mom would have to give Alice 50 cents.

8. Harrison's neighborhood can be plotted on a Cartesian plane. The axes mark different boundaries of different properties. Each quadrant is one neighbor's property. One of the neighbors, Chris, is very territorial and will punish anyone who trespasses. One day Harrison's friends dare him to enter Chris' property. Although Harrison doesn't want to do it, he also doesn't want to be made fun of for being scared. Instead, he decides to walk in a path that makes it seem like he will enter the property, but will actually never do it (basically an asymptote). His path can be represented on the Cartesian plane as: y = b^x when 0 < b < 1 In which quadrant is Chris' property?

Answer: IV

Since the base of the function has to be a fraction, the asymptote of the function is in the first quadrant. The higher the value of x, the smaller value of y. No matter how high x will go, y will never reach zero. In Harrison's case, he will continue to walk along the boundary into his neighbor's property, but will never cross it. Since the asymptote is following the x axis in the first quadrant, the property is in the fourth quadrant.

9. Jackie is opening up a business, and she contacts her financial adviser. Jackie is wondering if she has enough funds to start a successful business. Jackie has "x" dollars for the equation x^3 = 100^6. How much money does she have?

Answer: $10000

To solve the equation, you need to get x by itself. You can do this by taking the cubed root of both sides.

x = (100^6)^(1/3)

If you multiply the exponents together you get 100^2, which equals 10000.

You can also solve the equation by using logarithms.

10. Jason's having trouble remembering the last number in his locker combination. Luckily for him he's written down the code, but in a mathematical form. Starting at n=2, find the product of all the values of 9^(1/2^n) as n increases by one integer to infinity. In other words, find the product of (9^(1/2^2))*(9^(1/2^3))*(9^(1/2^4))*(9^(1/2^5))*... The product is the last number of Jason's code. So, what is his last code number?

Answer: 3

You should first plug in n for the first couple of terms. For n=2, the term is 9^(1/4). For n=3, it is 9^(1/8). For n=4, it is 9^(1/16). Do you notice a pattern in the exponents? Each succeeding term has its exponent divided by 2. Since all the terms are being multiplied together and have the same base, you add all of the exponents:

9^((1/4)+(1/8)+(1/16)+(1/32)+(1/64)+...)

This is an infinite geometric series. The formula used to find the sum of one is:

a/(1-r)

"a" is the first term in the series, and "r" is the ratio at which each succeeding term is multiplied by. Simply plug in the numbers to find the sum:

(1/4)/(1 - (1/2))

It comes out to be 1/2. So, the sum of all the infinite number of exponents equals 1/2. And 9 to the 1/2 power is 3. So, Jason's last locker code is 3.

Source: Author dijonmustard

This quiz was reviewed by FunTrivia editor crisw before going online.
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The Power of Exponents Trivia Quiz

Besides being a topic in math class, exponents are powerful tools in the real world. Take this quiz to see how much you know of exponents...preferably without a calculator.

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