Q67. Conjecture The square of an inte... [FREE SOLUTION]

91影视

Pre-algebra

Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff

$Math Studyset 91影视 Explanations$ Math

Common Core Edition 路 183 pages

ISBN: 9780547587776

Chapter 4: Q67. (page 174)

Conjecture The square of an integer is called a perfect square. Write the prime factorizations, with exponents, for these perfect squares: 4, 9,16, 25, 36, and 64. Make a conjecture about the exponents in the prime factorization of a perfect square.

Short Answer

Expert verified

A conjecture can be stated as 鈥渢he exponents in the prime factorization of a perfect square are all even鈥�.

Step by step solution

Define prime factorization

When a number is written as a product of prime numbers it is defined as prime factorization, wherein, prime numbers are defined as whole numbers that is greater than 1 and has exactly two whole number factors 1 and itself.

Write prime factorization of the perfect squares

Write the prime factorization of the following numbers: 4, 9, 16, 25, 36 and 64.

$\begin{array}{l} 4 = 2 脳 2 = 2^{2} \\ 9 = 3 脳 3 = 3^{2} \\ 16 = 4 脳 4 = 2 脳 2 脳 2 脳 2 = 2^{4} \\ 25 = 5 脳 5 = 5^{2} \\ 36 = 6 脳 6 = 3 脳 2 脳 3 脳 2 = 2^{2} 脳 3^{2} \\ 64 = 8 脳 8 = 2 脳 2 脳 2 脳 2 脳 2 脳 2 = 2^{6} \end{array}$

State the conclusion

Prime factorisation of the given numbers are:

$4 = 2^{2}, 9 = 3^{2}, 16 = 2^{4}, 25 = 5^{2}, 36 = 2^{2} 脳 3^{2} and 64 = 2^{6}$

From the prime factorization, it can be observed that all the perfect squares are even powers of the prime numbers. Therefore, a conjecture can be stated as 鈥渢he exponents in the prime factorization of a perfect square are all even鈥�.

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91影视

Conjecture The square of an integer is called a perfect square. Write the prime factorizations, with exponents, for these perfect squares: 4, 9,16, 25, 36, and 64. Make a conjecture about the exponents in the prime factorization of a perfect square.

Short Answer

Step by step solution

Define prime factorization

Write prime factorization of the perfect squares

State the conclusion

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