Depth · one concept from first principles · step 2 of 4
Rewrite in base 2 and factor the sum
When is simplified, it equals , where is a prime number and is a natural number. Find the value of .
Expressions
Your answer
How to solve
Strategy Organize Information in More Ways — 8 and 4 are both powers of 2, so write everything in base 2. The denominator then turns out to be a power of 2 times the numerator, and the fraction collapses.
1STEP 1
Write every power in base 2
Use 8=2³ and 4=2² with the power-of-a-power rule.
8⁴=2¹², 4⁴=2⁸, 8⁶=2¹⁸, 4⁷=2¹⁴
(2³)⁴=2^(3×4)=2¹², and so on.
8.EE.A.1Organize Information In More Ways2STEP 2
Factor the denominator
Each denominator term is 2⁶ times a numerator term.
2¹⁸+2¹⁴=2⁶×2¹²+2⁶×2⁸=2⁶(2¹²+2⁸)
This is the distributive property run backwards.
6.EE.A.3Identify Subproblems.
Why?
and , because multiplying powers of adds their counts of s.
🧱An exponent counts repeated multiplicationAn exponent records only how many times a factor is used, so multiplying two powers of the same base just adds those counts together.
Why?
Pulling the shared factor out of the sum is the distributive property read backwards.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
3STEP 3
Cancel the common sum
The same sum 2¹²+2⁸ appears on top and bottom.
(2¹²+2⁸)/(2⁶(2¹²+2⁸))=1/(2⁶)
Whatever the sum is, it divides out.
8.EE.A.1Identify Subproblems4STEP 4
Square the result
Square 1/(2⁶).
(1/(2⁶))²=1/(2¹²)=(1/2)¹²
Squaring doubles the exponent.
8.EE.A.1Convert To Algebra5STEP 5
Read off a and b
(1/a)^b=(1/2)¹² means a^b=2¹²; since a is prime it must be 2, so b=12.
b^a=12²=144
Only the prime 2 divides 2¹².
8.EE.A.1Convert To AlgebraAnswer
144
Check numerically: 8⁴+4⁴=4096+256=4352 and 8⁶+4⁷=262144+16384=278528. Then 278528÷4352=64=2⁶, so the fraction really is 1/(2⁶).
Takeaway
Exponent rules don't work on sums -- so factor the sum first, then the rules take over.
- Write every power in base 2
- Factor the denominator
- Cancel the common sum
- Square the result
- Read off a and b
Where next?
Another one like thissuggested
Step back and solidifysuggested
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