If \(96 - 64a^3 + \frac{8}{a^6} - \frac{48}{a^3 } - t^3 = 0\) then what is a 2t + 4a 3 equal to ?
2
Solving the Given Algebra Equation
The problem asks us to find the value of a specific expression involving variables \(a\) and \(t\), given a complex algebraic equation relating them.
The given equation is:
\[96 - 64a^3 + \frac{8}{a^6} - \frac{48}{a^3 } - t^3 = 0\]Let's rearrange the terms involving \(a\) to see if they form a recognizable pattern:
\[96 - 64a^3 - \frac{48}{a^3} + \frac{8}{a^6} = t^3\]Consider the cubic expansion of \((x-y)^3\):
\[(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3\]Let's examine the terms on the left side of the rearranged equation: \(96\), \(-64a^3\), \(-\frac{48}{a^3}\), \(\frac{8}{a^6}\). We notice that \(64a^3 = (4a)^3\) and \(\frac{8}{a^6} = (\frac{2}{a^2})^3\).
Let's try expanding \((\frac{2}{a^2} - 4a)^3\):
\[\left(\frac{2}{a^2} - 4a\right)^3 = \left(\frac{2}{a^2}\right)^3 - 3\left(\frac{2}{a^2}\right)^2(4a) + 3\left(\frac{2}{a^2}\right)(4a)^2 - (4a)^3\] \[= \frac{8}{a^6} - 3\left(\frac{4}{a^4}\right)(4a) + 3\left(\frac{2}{a^2}\right)(16a^2) - 64a^3\] \[= \frac{8}{a^6} - \frac{48a}{a^4} + \frac{96a^2}{a^2} - 64a^3\] \[= \frac{8}{a^6} - \frac{48}{a^3} + 96 - 64a^3\]This matches the terms involving \(a\) in the rearranged equation:
\[96 - 64a^3 - \frac{48}{a^3} + \frac{8}{a^6} = \left(\frac{2}{a^2} - 4a\right)^3\]So, the original equation can be written as:
\[\left(\frac{2}{a^2} - 4a\right)^3 = t^3\]Taking the cube root of both sides (considering real roots), we get a relation between \(t\) and \(a\):
\[\frac{2}{a^2} - 4a = t\]We can rewrite this relation by finding a common denominator:
\[t = \frac{2 - 4a^3}{a^2}\]Now, multiply both sides by \(a^2\) (assuming \(a \neq 0\)):
\[a^2 t = 2 - 4a^3\]Rearrange this equation to group the terms involving \(a^3\) and \(a^2 t\):
\[a^2 t + 4a^3 = 2\]This equation shows that the expression \(a^2 t + 4a^3\) is equal to a constant value, 2.
Evaluating the Expression
The question asks for the value of the expression "a 2t + 4a 3". Based on standard mathematical notation and the common structure of such problems where the answer is a constant, the phrasing "a 2t" is most likely intended to represent \(a^2 t\), and "4a 3" represents \(4a^3\). Thus, the expression to be evaluated is likely \(a^2 t + 4a^3\).
From our derivation based on the given equation, we found that:
\[a^2 t + 4a^3 = 2\]Therefore, the value of the expression \(a^2 t + 4a^3\) is 2.
If, alternatively, the expression was intended to be \(a(2t) + 4a^3 = 2at + 4a^3\), substituting \(t = \frac{2}{a^2} - 4a\) gives \(2a(\frac{2}{a^2} - 4a) + 4a^3 = \frac{4}{a} - 8a^2 + 4a^3\), which is not a constant for all valid \(a\). However, our derivation \(a^2 t + 4a^3 = 2\) is directly obtained from the original equation and involves the terms \(a^2t\) and \(4a^3\). Given the options are constants, it is highly probable that the intended expression was \(a^2 t + 4a^3\).
Let's verify the options. The options are 0, 1, 2, 3. Our derived value is 2, which is one of the options.
The final answer is obtained by simplifying the relation derived from the equation.
Step-by-Step Solution Summary
- Start with the given equation: \(96 - 64a^3 + \frac{8}{a^6} - \frac{48}{a^3 } - t^3 = 0\).
- Rearrange the terms involving \(a\): \(96 - 64a^3 - \frac{48}{a^3} + \frac{8}{a^6} = t^3\).
- Recognize the left side as the cubic expansion: \((\frac{2}{a^2} - 4a)^3\).
- Rewrite the equation: \((\frac{2}{a^2} - 4a)^3 = t^3\).
- Take the cube root: \(\frac{2}{a^2} - 4a = t\).
- Rewrite the relation with a common denominator: \(t = \frac{2 - 4a^3}{a^2}\).
- Multiply by \(a^2\): \(a^2 t = 2 - 4a^3\).
- Rearrange to find the value of \(a^2 t + 4a^3\): \(a^2 t + 4a^3 = 2\).
- Assuming the expression "a 2t + 4a 3" refers to \(a^2 t + 4a^3\), its value is 2.
This value matches one of the given options.
Revision Table: Key Concepts
| Concept | Description | Application in Solution |
|---|---|---|
| Cubic Expansion | Formula: \((x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3\) | Used to simplify the left side of the equation. |
| Algebraic Manipulation | Rearranging terms, finding common denominators, multiplying equations. | Used throughout the solution to simplify the equation and the relation. |
| Cube Root | Inverse operation of cubing. | Applied to solve for \(t\) from \((\frac{2}{a^2} - 4a)^3 = t^3\). |
| Equation Solving | Finding the relationship between variables that satisfies the equation. | The given equation leads to the relation \(a^2 t + 4a^3 = 2\). |
Additional Information: Recognizing Algebraic Patterns
In complex algebraic problems, recognizing patterns like binomial expansions (e.g., \((x \pm y)^2\), \((x \pm y)^3\)) is crucial. The terms in the equation \(96 - 64a^3 + \frac{8}{a^6} - \frac{48}{a^3 }\) involve \(a^3\), \(1/a^3\), \(1/a^6\), and a constant. These forms strongly suggest a cubic expansion involving terms like \(a\) and \(1/a^2\) or \(1/a^3\). By carefully matching the coefficients and powers in the given equation with the terms generated by expanding \((X-Y)^3\), we can identify the specific binomial (\(\frac{2}{a^2} - 4a\)) that simplifies the expression.
Another important technique used here is rearranging equations to isolate variables or specific expressions. By rearranging the derived relation \(t = \frac{2}{a^2} - 4a\) into the form \(a^2 t + 4a^3 = 2\), we directly obtain the value of the expression requested (assuming the interpretation \(a^2 t + 4a^3\)). This highlights how solving the underlying relationship between variables is key to evaluating expressions involving them.
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