If (2^{x}\cdot8^{x-2}=64), what is the value of (x)?
Answer and explanation
Correct answer: (3)
The key step is to express every factor with the same base, 2. Since \\(8=2^3\\) and \\(64=2^6\\), the equation can be compared using powers of 2. The exponent on the left becomes \\(4x-6\\). Equating exponents gives \\(x=3\\), so option B is correct.
Rewrite the equation as \\(2^x(2^3)^{x-2}=2^6\\). Using the power rule, \\((2^3)^{x-2}=2^{3x-6}\\), and multiplying like bases gives \\(2^{x+3x-6}=2^{4x-6}\\). Therefore \\(4x-6=6\\), so \\(4x=12\\) and \\(x=3\\). Substitution checks it: \\(2^3\\cdot8^1=8\\cdot8=64\\).
Frequently asked questions
What is the correct answer to this question?
(3)
Why is this the correct answer?
The key step is to express every factor with the same base, 2. Since \\(8=2^3\\) and \\(64=2^6\\), the equation can be compared using powers of 2. The exponent on the left becomes \\(4x-6\\). Equating exponents gives \\(x=3\\), so option B is correct.
Rewrite the equation as \\(2^x(2^3)^{x-2}=2^6\\). Using the power rule, \\((2^3)^{x-2}=2^{3x-6}\\), and multiplying like bases gives \\(2^{x+3x-6}=2^{4x-6}\\). Therefore \\(4x-6=6\\), so \\(4x=12\\) and \\(x=3\\). Substitution checks it: \\(2^3\\cdot8^1=8\\cdot8=64\\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Operations on real numbers and the laws of exponents.
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