To determine which of the given expressions cannot be written in the form of \(4^a x\) where \(a\) is an integer, let's simplify each expression one by one.
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\(\frac{4^3}{4} x\): \[ \frac{4^3}{4} = 4^{3-1} = 4^2 \] This can be rewritten as \(4^2 x\) where \(a = 2\).
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\((2 \times 4) x\): \[ 2 \times 4 = 8 = 4^{3/2} \] This cannot be expressed as \(4^a x\) where \(a\) is an integer (since \(3/2\) is not an integer).
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\(2 \times x^2\) (assuming that "second power" refers to \(x^2\)): This can be expressed as \(2 x^2\). Since \(2\) cannot be written in the form of \(4^a\), it is not expressible as \(4^a x\).
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\(\frac{4^5 x}{4^2}\): \[ \frac{4^5}{4^2} x = 4^{5-2} x = 4^3 x \] This can be rewritten as \(4^3 x\) where \(a = 3\).
Now, summarizing:
- \(\frac{4^3}{4} x = 4^2 x\) is valid.
- \((2 \times 4) x = 8 x\) is not valid in the form \(4^a x\).
- \(2 x^2\) is not valid in the form \(4^a x\) since \(2\) can't be expressed as \(4^a\).
- \(\frac{4^5 x}{4^2} = 4^3 x\) is valid.
Result: The expressions that cannot be written in the form of \(4^a x\) are \((2 \times 4) x\) and \(2 x^2\). However, if selecting one option, both \(2 \times 4\) and \(2 x^2\) cannot fit, but \(2 \times 4\) is the first clear invalid instance you provided.