Lesson 8
Combining Bases
Problem 1
Select all the true statements:
\(2^8 \boldcdot 2^9 = 2^{17}\)
\(8^2 \boldcdot 9^2 = 72^2\)
\(8^2 \boldcdot 9^2 = 72^4\)
\(2^8 \boldcdot 2^9 = 4^{17}\)
Solution
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Problem 2
Find \(x\), \(y\), and \(z\) if \((3 \boldcdot 5)^4 \boldcdot (2 \boldcdot 3)^5 \boldcdot (2 \boldcdot 5)^7 = 2^x \boldcdot 3^y \boldcdot 5^z\).
Solution
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Problem 3
Han found a way to compute complicated expressions more easily. Since \(2 \boldcdot 5 = 10\), he looks for pairings of 2s and 5s that he knows equal 10. For example, \(3 \boldcdot 2^4 \boldcdot 5^5 = 3 \boldcdot 2^4 \boldcdot 5^4 \boldcdot 5 = (3 \boldcdot 5) \boldcdot (2 \boldcdot 5)^4 = 15 \boldcdot 10^4 = 150,\!000.\) Use Han's technique to compute the following:
- \(2^4 \boldcdot 5 \boldcdot (3 \boldcdot 5)^3\)
- \(\frac{2^3 \boldcdot 5^2 \boldcdot (2 \boldcdot 3)^2 \boldcdot (3 \boldcdot 5)^2}{3^2}\)
Solution
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Problem 4
The cost of cheese at three stores is a function of the weight of the cheese. The cheese is not prepackaged, so a customer can buy any amount of cheese.
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Store A sells the cheese for \(a\) dollars per pound.
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Store B sells the same cheese for \(b\) dollars per pound and a customer has a coupon for $5 off the total purchase at that store.
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Store C is an online store, selling the same cheese at \(c\) dollar per pound, but with a $10 delivery fee.
This graph shows the price functions for stores A, B, and C.
![Coordinate plane, horizontal, weight of cheese in pounds, 0 to 12 by 2, vertical, cost in dollars,](https://cms-im.s3.amazonaws.com/hE75bmmZKquehgeKxSjBTLbQ?response-content-disposition=inline%3B%20filename%3D%228-8.5.C.PP.Image.16.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.C.PP.Image.16.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T182345Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2a9d8958a490c75f444c5b18d6c9c6094cc840cbce2b3676e1b88092073fb073)
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Match Stores A, B, and C with Graphs \(j\), \(k\), and \(\ell\).
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How much does each store charge for the cheese per pound?
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How many pounds of cheese does the coupon for Store B pay for?
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Which store has the lowest price for a half a pound of cheese?
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If a customer wants to buy 5 pounds of cheese for a party, which store has the lowest price?
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How many pounds would a customer need to order to make Store C a good option?
Solution
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(From Unit 5, Lesson 8.)