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 values that make the equation true are 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. We have a few options for a and b , and we need to determine which pair satisfies the equation.
Eliminating Square Roots First, let's square both sides of the equation to eliminate the square roots. This gives us 648 = 2 a ⋅ 3 b . Now we need to find the prime factorization of 648 to express it as a product of powers of 2 and 3.
Prime Factorization of 648 To find the prime factorization of 648, we can use the division method. We start by dividing 648 by the smallest prime number, 2: 648 ÷ 2 = 324 324 ÷ 2 = 162 162 ÷ 2 = 81 So, 648 = 2 3 ⋅ 81 . Now we need to factor 81. Since 81 is not divisible by 2, we try the next smallest prime number, 3: 81 ÷ 3 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 So, 81 = 3 4 . Therefore, the prime factorization of 648 is 648 = 2 3 ⋅ 3 4 .
Finding a and b Now we can compare the prime factorization of 648 with the equation 648 = 2 a ⋅ 3 b . We have 2 3 ⋅ 3 4 = 2 a ⋅ 3 b . By comparing the exponents of 2 and 3, we can see that a = 3 and b = 4 .
Final Answer Therefore, the values of a and b that make the equation true are a = 3 and b = 4 .
Conclusion The values of a and b that satisfy the equation 648 = 2 a ⋅ 3 b are a = 3 and b = 4 .
Examples
Prime factorization is a fundamental concept in number theory with many 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. In music, understanding prime factors can help explain why certain musical intervals sound harmonious. For instance, the octave (2:1) and the perfect fifth (3:2) have simple prime ratios, which contribute to their consonance.
The values of a and b that satisfy the equation 648 = 2 a ⋅ 3 b are a = 3 and b = 4 . Therefore, the correct option is C: a = 3 , b = 4 .
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