What values of $a$ and $b$ make the equation true?
[tex]\sqrt{648}=\sqrt{2^a \cdot 3^b}[/tex]
A. $a=3, b=2$
B. $a=2, b=3$
C. $a=3, b=4$
D. $a=4, b=3$
Answer (2)
Square both sides of the equation: 648 = 2 a c d o t 3 b .
Find the prime factorization of 648: 648 = 2 3 c d o t 3 4 .
Compare the exponents: a = 3 and b = 4 .
The solution is a = 3 , b = 4 .
Explanation
Problem Analysis We are given the equation 648 = 2 a ⋅ 3 b and asked to find the values of a and b that make the equation true. The possible answers are: a = 3 , b = 2 a = 2 , b = 3 a = 3 , b = 4 a = 4 , b = 3
Squaring Both Sides To solve this, we first square both sides of the equation to get rid of the square roots: ( 648 ) 2 = ( 2 a ⋅ 3 b ) 2 648 = 2 a ⋅ 3 b
Prime Factorization of 648 Now we need to find the prime factorization of 648. From the tool, we know that the prime factorization of 648 is 2 3 ⋅ 3 4 .
Comparing Exponents Comparing this with the equation 648 = 2 a ⋅ 3 b , we can see that a = 3 and b = 4 .
Finding the Values of a and b Therefore, the correct values are a = 3 and b = 4 .
Final Answer Thus, the values of a and b that make the equation true are a = 3 and b = 4 .
Examples
Prime factorization is a fundamental concept in number theory with various real-world applications. For example, in cryptography, the security of many encryption algorithms relies on the difficulty of factoring large numbers into their prime factors. Another application is in simplifying fractions or finding the least common multiple (LCM) or greatest common divisor (GCD) of two numbers. Understanding prime factorization helps in many areas of mathematics and computer science.
The values of a and b that satisfy the equation are a = 3 and b = 4 . Therefore, the correct choice is C. a = 3 , b = 4 .
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