Simplify. Assume all variables are positive.
[tex]x^{-\frac{5}{2}} \div x^{-\frac{8}{7}}[/tex]
Write your answer in the form [tex]A[/tex] or [tex]\frac{A}{B}[/tex], where [tex]A[/tex] and [tex]B[/tex] are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.
Answer (2)
Rewrite the division as multiplication by the reciprocal: x − 2 5 ÷ x − 7 8 = x − 2 5 ⋅ x 7 8 .
Combine the exponents: x − 2 5 ⋅ x 7 8 = x − 2 5 + 7 8 .
Simplify the exponent: − 2 5 + 7 8 = − 14 19 .
Rewrite with a positive exponent: x − 14 19 = x 14 19 1 .
The final answer is x 14 19 1 .
Explanation
Understanding the Problem We are given the expression x − 2 5 ÷ x − 7 8 . Our goal is to simplify this expression, assuming that x is a positive variable. We need to express the final answer in the form A or B A , where A and B are constants or variable expressions with no variables in common, and all exponents should be positive.
Rewriting the Division First, let's rewrite the division as multiplication by the reciprocal. Recall that dividing by a term with a negative exponent is the same as multiplying by the same term with a positive exponent. So, we have: x − 2 5 ÷ x − 7 8 = x − 2 5 ⋅ x 7 8
Combining Exponents Now, we use the property of exponents that states x a ⋅ x b = x a + b . Applying this property, we get: x − 2 5 ⋅ x 7 8 = x − 2 5 + 7 8
Finding a Common Denominator To add the exponents, we need to find a common denominator for the fractions − 2 5 and 7 8 . The least common denominator for 2 and 7 is 14. So, we rewrite the fractions with the common denominator: − 2 5 + 7 8 = − 2 ⋅ 7 5 ⋅ 7 + 7 ⋅ 2 8 ⋅ 2 = − 14 35 + 14 16
Simplifying the Exponent Now, we add the fractions: − 14 35 + 14 16 = 14 − 35 + 16 = 14 − 19 So, the exponent is − 14 19 .
Substituting the Exponent Substitute the simplified exponent back into the expression: x − 14 19
Writing with Positive Exponent Since we want to express the answer with a positive exponent, we use the property x − a = x a 1 :
x − 14 19 = x 14 19 1 This is in the form B A , where A = 1 and B = x 14 19 . Since A and B have no variables in common, and the exponent is positive, this is the desired form.
Final Answer Therefore, the simplified expression is x 14 19 1 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, including physics and engineering. For example, when dealing with wave phenomena, such as light or sound, the intensity of the wave often depends on the distance from the source raised to a negative power. Simplifying such expressions helps in calculating and predicting wave behavior. Imagine you're designing a solar panel. The amount of sunlight that reaches the panel decreases with the square of the distance from the sun. By simplifying exponential expressions, you can accurately calculate the panel's efficiency based on its distance from the sun, ensuring optimal energy capture.
To simplify x − 2 5 ÷ x − 7 8 , we rewrite it as multiplication by the reciprocal, combine the exponents, and simplify them. The final answer is x 14 19 1 , which has positive exponents. This form meets the specified requirements of the problem.
;
