\frac{-5 w^4 y^{-2}}{-15 w^{-6} y^2}, w \neq 0, y \neq 0
$\frac{1}{3 w ^{-2}}$
$\frac{1}{3 y^4 w^2}$
$\frac{w^{10}}{3 y^4}$
$\frac{w^2}{3 y^4}$
Answer (2)
3 y 4 w 10
Explanation
Understanding the Problem We are asked to simplify the expression − 15 w − 6 y 2 − 5 w 4 y − 2 , given that w e q 0 and ye q 0 . This means that neither w nor y can be equal to zero. We will simplify the expression by dividing the coefficients and using the quotient rule for exponents.
Simplifying Coefficients First, let's simplify the fraction by dividing the coefficients: − 15 − 5 = 3 1
Simplifying w terms Next, we use the quotient rule for exponents to simplify the terms with w : w − 6 w 4 = w 4 − ( − 6 ) = w 4 + 6 = w 10
Simplifying y terms Now, we use the quotient rule for exponents to simplify the terms with y : y 2 y − 2 = y − 2 − 2 = y − 4
Combining and Rewriting Combining the simplified terms, we have: 3 1 w 10 y − 4 To rewrite the expression with positive exponents, we move y − 4 to the denominator: 3 y 4 w 10
Examples
In physics, when dealing with wave equations or electromagnetic fields, you often encounter expressions with exponents and fractions. Simplifying these expressions, like the one we just did, helps in understanding the relationships between different physical quantities. For example, if w represents frequency and y represents wavelength, the simplified expression could relate these to energy density or intensity of the wave. This kind of simplification is crucial for making calculations and predictions about wave behavior.
The simplified expression for − 15 w − 6 y 2 − 5 w 4 y − 2 is 3 y 4 w 10 . This is obtained by simplifying both the coefficients and the variables using the rules for exponents. Thus, the final answer is 3 y 4 w 10 .
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