AMC 8 2013 Problem 15

AMC8 2013 年第 15 题

表达式与运算★★★☆☆

3^p + 3^4 = 90

2^r + 44 = 76

, and

5^3 + 6^s = 1421

, what is the product of

p

r

, and

s

若

3^p + 3^4 = 90

，

2^r + 44 = 76

，

5^3 + 6^s = 1421

，那么

p

、

r

、

s

的乘积是多少？

正确答案 · Correct Answer

解析 · Solution

Solution 1

: Solving First, we're going to solve for

p

. Start with

3^p+3^4=90

. Then, change

3^4

to 81. Subtract 81 from both sides to get

3^p=9

and see that

p

is 2. Now, solve for

r

. Since

2^r+44=76

2^r

must equal 32, so r=5. Now, solve for

s

5^3+6^s=1421

can be simplified to

125+6^s=1421

which simplifies further to

6^s=1296

. Therefore, s=4. $prs$ equals 2*5*4 which equals 40. So, the answer is 40.: Process of Elimination First, we solve for

s

. As perfectly states,

5^3+6^s=1421

can be simplified to

125+6^s=1421

which simplifies further to

6^s=1296

. Therefore, s=4. We know that you cannot take a root of any of the numbers raised to

p

r

, or

s

and get a rational answer, and none of the answer choices are irrational, so that rules out the possibility that

p

r

, or

s

is a fraction. The only answer choice that is divisible by 4 is 40.

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