Math Problem Statement
Solution
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Transcription of the Questions:
1. Perform the calculations (simplify if possible):
(a)
(b)
(c)
(d)
(e)
2. Compare and :
3. Compare and :
4. Find all prime numbers such that and are also primes.
Solutions:
1. Perform the calculations:
(a)
(b) Simplify the fractions:
= \frac{3^6 \cdot 5^6}{2^2 \cdot 3^2 \cdot 5^6} = \frac{3^4}{2^2} = \frac{81}{4}.$$ --- (c) Simplify powers: $$\frac{(3 \cdot 4 \cdot 2^{16})^2}{11 \cdot 12^{13} \cdot 4^{11} \cdot 16^9} = \frac{(3 \cdot 4 \cdot 2^{16})^2}{11 \cdot (3 \cdot 4)^{13} \cdot (2^2)^{11} \cdot (2^4)^9}.$$ Expand and simplify: $$= \frac{3^2 \cdot 4^2 \cdot 2^{32}}{11 \cdot 3^{13} \cdot 4^{13} \cdot 2^{22}} = \frac{2^{32 - 22} \cdot 3^{2 - 13} \cdot 4^{2 - 13}}{11}.$$ --- Would you like me to continue solving all parts step by step, or should I focus on any specific section?Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Arithmetic operations
Exponentiation
Fraction simplification
Number theory
Prime numbers
Formulas
Fraction simplification rules: a^m * a^n = a^(m+n), (a/b)^n = a^n / b^n
Sum of geometric series: S = a(1 - r^n) / (1 - r), for |r| < 1
Theorems
Properties of exponents
Prime number definition
Suitable Grade Level
Grades 9-12
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