geno3141

To get an x³ in the answer:

---   Multiply the +1 from the first factor with the 2x³ of the second factor to get 2x³

---   Multiply the -8x times the -7x² to get 56x³

---   Multiply the -3x² times the 3x to get -9x³

---   Multiply the -3x³ times the 4 to get -12x³

Add these terms together to get the coefficient that you want ...

Write  4^x as  (2²)^x  =  2^2x

Then  4^x - 2^x = 56  becomes  2^2x - 2^x = 56   which is:   2^2x - 2^x - 56  =  0

Now, factor:  (2^x - 8)(2^x + 7)  =  0

Either:  2^x - 8  =  0     --->   2^x  =  8     --->     x  =  3

Or:       2^x + 7  =  0     --->   2^x  =  -7    (impossible for real numbers)

If the problem is:  5log₅(2v + 4) - 4  =  -9

Add 4 to both sides:     5log₅(2v + 4)  =  -5

Divide both sides by 5:  log₅(2v + 4)  =  -1      

Write into exponential notation:  2v + 4  =  5^-1    --->    2v + 4  =  1/5

Subtract 4 from both sides:        2v  =  -3.8

Divide both sides by 2:                v  =  -1.9 

Let's see:  you start with 84 quarters.

Now you change every 2nd quarter with a dime; so you now have 42 quarters (those in odd-numbered locations) and 42 dimes (those in even numbered locations).

When you change every third coin with a nickel, half of them will come from odd-numbered locations and half will come from even-numbered locations. Since 84 divided by 3 is 28, 14 nickels will be in odd-numbered locations and 14 will be in even numbered locations. You now have 28 quarters and 14 nickels in odd-numbered locations and 28 dimes and 14 nickels in even-numbered locations.

Changing every fourth coin with a penny, will place 21 pennies, all in even-numbered locations; 14 in locations that have dimes and 7 in locations that have nickels.

In odd-numbered locations, you have 28 quarters and 14 nickels; in even-numbered locations, you have 14 dimes, 14 pennies, 7 nickels and 7 pennies.

Total:  28 quarters, 14 dimes, 21 nickels, and 21 pennies

Picture a right triangle with the right angle being ∠C. Label the other two angles ∠A and ∠B.

cos is defined to be the ratio of the adjacent side to the hypotenuse.

In this triangle, AB is the hypotenuse, side AC is the side adjacent to ∠A and side BC is the side opposite to ∠A.

cos∠A  =  3/5  =  AC/AB

Since this is a right triangle, use the Pythagorean Theorem to find the value of the third side:

   AC² + BC²  =  AB²     --->     3² + BC²  =  5²      --->   BC  =  4

sin is defined to be the ratio of the opposite side to the hypotenuse.

sin∠A  =  BC/AB  =  4/5

tan is defined to be the ratio of the opposite side to the adjacent side.

tan∠A  =  BC/AC  =  4/3

log₁₀N = 4  can be written as  10⁴ = N   --->   N  =  10 000

Drawing three cards and then returning these cards and reshuffling does not influence the rest of the problem.

The problem of getting three spades by drawing three cards:

  You must get a spade on the first card: the probability is 13/52 because there are 13 spades and 52 cards

    and you must get a spade on the next card:  12/51 because there are 12 spades left and 51 cards left

      and you must get a spade on the third card:  11/50  because there are 11 spades left and 50 cards left

  The total probability is 13/52 x 12/51 x 10/50  (you multiply when you have an 'and' situation).

The problem of getting a ten, a king, and a two (in that order):

  You must get a ten on the first card:  4/52  (four tens and 52 cards)

    and you must get a king on the second card:  4/51  (four kings and 51 remaining cards)

      and you must get a two on the third card:  4/50  (four twos and 50 remaining cards)

  The total probability is 4/52 x 4/51 x 4/50

If you mean 1/4%:

1/4% means 1/4 of 1%

which is 1/4 of 0.01

which is 1/4 x 0.01

which is 0.25 x 0.01   =   0.0025

Using the 50 fish that were caught to represent all the fish in the lake, we can set up this proportion:

14 / 50  =  x / 7500

Can you finish this?

To be 0.04 from 0.92, you could be 0.04 smaller than 0.92 or 0.04 larger than 0.92.

If you are 0.04 smaller than 0.92, you are 0.92 - 0.04  = 0.88.

If you are 0.04 larger than 0.92, you are 0.92 + 0.04  =  0.96.

Both of these numbers are larger than 0.75 and smaller than 1.20.