Which of these best explains the next step to simplify this expression?
[tex]\frac{3 y^{-4}}{5}=\frac{3}{5 y^{4}}[/tex]
A. Multiply 5 by -4.
B. Keep the exponent in the denominator -4.
C. Make the -4 exponent in the denominator positive.
D. Add 5 to -4.
Answer (2)
Recognize that a negative exponent can be moved from the numerator to the denominator (or vice versa) by changing its sign.
Identify the term with the negative exponent in the denominator: y − 4 .
Make the exponent positive by moving the term to the numerator: y 4 .
Therefore, the next step is to make the -4 exponent in the denominator positive.
Explanation
Understanding the Expression We are given the expression 5 3 y − 4 = 5 y − 4 3 and asked to identify the best next step to simplify it.
Understanding Negative Exponents The key to simplifying this expression lies in understanding how to handle negative exponents. Recall that a − n = a n 1 . This means a term with a negative exponent in the numerator can be moved to the denominator by changing the sign of the exponent, and vice versa.
Moving the Term with the Negative Exponent In the given expression, we have y − 4 in the numerator. To simplify, we can move this term to the denominator and change the sign of the exponent from -4 to +4. This gives us 5 y 4 3 .
Simplifying the Given Expression However, the expression is already written as 5 y − 4 3 . To simplify this, we need to move y − 4 from the denominator to the numerator and change the sign of the exponent. This would give us 5 3 y 4 . The question asks for the next step to simplify the expression 5 3 y − 4 = 5 y − 4 3 . The right-hand side has y − 4 in the denominator. To simplify this, we need to make the -4 exponent in the denominator positive by moving y − 4 to the numerator.
Conclusion Therefore, the best next step is to make the -4 exponent in the denominator positive.
Examples
Understanding how to simplify expressions with negative exponents is crucial in various fields, such as physics and engineering. For example, in physics, the gravitational force between two objects is inversely proportional to the square of the distance between them, which can be expressed using a negative exponent. Simplifying such expressions allows for easier calculations and a better understanding of the relationships between variables. In finance, understanding exponential growth and decay is essential for calculating investments and loans. Simplifying expressions with negative exponents helps in accurately determining the present and future values of assets.
The next step to simplify the expression 5 3 y − 4 is to make the -4 exponent in the denominator positive by moving it to the numerator. This will transform the expression into 5 y 4 3 . Thus, the correct answer is option C.
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