Simplify the expression:
[tex]\begin{array}{c}
\left(r^{-4} s^7 t^5\right)^{10} \\
\frac{s^{[?]} t}{r}
\end{array}[/tex]
Answer (2)
Simplify the numerator using the power of a product rule: ( r − 4 s 7 t 5 ) 10 = r − 40 s 70 t 50 .
Multiply the numerator by the reciprocal of the denominator: r s [ ?] t r − 40 s 70 t 50 = r − 40 s 70 t 50 ⋅ s [ ?] t r .
Combine terms with the same base by adding exponents: r − 40 + 1 s 70 − [ ?] t 50 − 1 = r − 39 s 70 − [ ?] t 49 .
The simplified expression is r − 39 s 70 − [ ?] t 49 .
Explanation
Understanding the Problem We are given the expression r s [ ?] t ( r − 4 s 7 t 5 ) 10 and we need to simplify it and find the missing exponent of s in the denominator.
Simplifying the Numerator First, let's simplify the numerator by applying the power of a product rule: ( r − 4 s 7 t 5 ) 10 = r − 4 × 10 s 7 × 10 t 5 × 10 = r − 40 s 70 t 50 So the expression becomes r s [ ?] t r − 40 s 70 t 50
Simplifying the Expression Now, let's simplify the entire expression by multiplying the numerator by the reciprocal of the denominator: r s [ ?] t r − 40 s 70 t 50 = r − 40 s 70 t 50 ⋅ s [ ?] t r = r − 40 s 70 t 50 r 1 s − [ ?] t − 1 Combining the terms with the same base by adding the exponents, we get: r − 40 + 1 s 70 − [ ?] t 50 − 1 = r − 39 s 70 − [ ?] t 49 So the expression simplifies to r − 39 s 70 − [ ?] t 49
Finding the Missing Exponent We want to find the exponent of s such that the simplified expression is r s [ ?] t However, our simplified expression is r − 39 s 70 − [ ?] t 49 To match the form r s [ ?] t , we can rewrite it as r − 1 s [ ?] t Comparing the exponents of r and t , we have: r − 39 = r − 1 ⟹ − 39 = − 1 This is not possible, so there must be a typo in the problem. Assuming the simplified expression should be in the form r − 39 s x t 49 , we can proceed to find the value of x .
We have r − 39 s 70 − [ ?] t 49 . Comparing this to the given form, we need to find the value of [?] such that the expression is simplified. The problem asks us to simplify the expression, so we need to find the value of [?] such that the expression is in its simplest form. In this case, we want to find the value of [?] such that the expression is in the form r − 39 s 70 − [ ?] t 49 .
If the question is asking what value should replace the question mark to simplify the expression, we can leave the answer as r − 39 s 70 − [ ?] t 49 . However, if the question is asking for a specific value of the exponent of s , we need more information.
Possible Interpretations Let's assume the problem meant to ask what value of the exponent of s in the original expression would result in the simplified expression having no s term. In that case, we would need 70 − [ ?] = 0 , which means [ ?] = 70 . Then the expression would simplify to r − 39 t 49 .
However, without further information or clarification, we can only simplify the expression to r − 39 s 70 − [ ?] t 49 .
Final Answer Based on the original problem statement, we have simplified the expression to r − 39 s 70 − [ ?] t 49 . Without additional information, we cannot determine a specific numerical value for the missing exponent. However, if we assume that the exponent of s in the simplified expression is 0, then 70 − [ ?] = 0 , so [ ?] = 70 .
Examples
When simplifying complex algebraic expressions, understanding exponent rules is crucial. For example, in physics, when dealing with quantities like force or energy that involve multiple variables raised to different powers, simplifying the expression helps in easier calculations and interpretations. Consider the expression for kinetic energy, K E = 2 1 m v 2 . If we have multiple objects with different masses and velocities, simplifying the expression before plugging in the values makes the calculation more manageable. Similarly, in circuit analysis, simplifying expressions involving resistance, current, and voltage helps in designing efficient circuits.
To simplify the expression, we found that [ ?] = 70 to achieve the final expression structure. The simplification involves applying exponent rules and comparing terms. Ultimately, the simplified expression becomes r − 39 t 49 .
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