Simplify. Express the answers using positive exponents. [tex]\frac{-2 a^2 b^4}{4 a b^{-8}}, a \neq 0, b \neq 0[/tex]
Answer (2)
Simplify the numerical coefficients: 4 − 2 = − 2 1 .
Simplify the 'a' terms: a a 2 = a .
Simplify the 'b' terms: b − 8 b 4 = b 12 .
Combine the simplified terms: − 2 1 a b 12 . The final answer is − 2 1 a b 12 .
Explanation
Understanding the Problem We are asked to simplify the expression 4 a b − 8 − 2 a 2 b 4 and express the answer using positive exponents, given that a e q 0 and b e q 0 .
Simplifying Coefficients First, let's simplify the numerical coefficients: 4 − 2 = − 2 1 .
Simplifying 'a' terms Next, we simplify the terms with the variable a . Using the quotient rule for exponents, we have a a 2 = a 2 − 1 = a 1 = a .
Simplifying 'b' terms Now, we simplify the terms with the variable b . Using the quotient rule for exponents, we have b − 8 b 4 = b 4 − ( − 8 ) = b 4 + 8 = b 12 .
Combining Terms Finally, we combine all the simplified terms: − 2 1 × a × b 12 = − 2 1 a b 12 .
Final Answer Therefore, the simplified expression is − 2 1 a b 12 .
Examples
In physics, when dealing with quantities that involve ratios of variables, such as calculating the density of a material ( ρ = V m ), simplifying expressions with exponents becomes crucial. For instance, if you have a complex formula involving ratios of masses and volumes with different exponents, simplifying it using exponent rules helps in easier computation and better understanding of the relationship between the variables. This is also applicable in other fields like finance, where rates and growth factors are often expressed with exponents.
The expression 4 a b − 8 − 2 a 2 b 4 simplifies to − 2 1 a b 12 by simplifying the numerical coefficients and using the rules for exponents. Each step involved reducing the coefficients and applying the properties of exponents to the variables. The result is expressed using positive exponents as required.
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