d. 𝑦 = √(𝑥3 − 4𝑥)2 3 e. 4𝑥7𝑦6 𝑦2

Let's address each question step by step.

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Part d: Simplify $y = \sqrt{(x^3 - 4x)^2}$

The square root of a square can be simplified as:

\sqrt{a^2} = |a| \quad \text{(absolute value of a)}.

Thus:

$y = \sqrt{(x^3 - 4x)^2} = |x^3 - 4x|.$

Final Simplified Form:

$y = |x^3 - 4x|.$

This expression ensures that the result is always non-negative.

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Part e: Simplify $\frac{4x^7 y^6}{y^2} = \frac{3x^5 y^4}{x^3}$

To simplify, let's break it down step by step:

Left-Hand Side:

$\frac{4x^7 y^6}{y^2}$

• Simplify the $y$-terms: $y^6 \div y^2 = y^{6-2} = y^4$.
• The $x$-terms remain as is.

$\frac{4x^7 y^6}{y^2} = 4x^7 y^4.$

Right-Hand Side:

$\frac{3x^5 y^4}{x^3}$

• Simplify the $x$-terms: $x^5 \div x^3 = x^{5-3} = x^2$.
• The $y$-terms remain as is.

$\frac{3x^5 y^4}{x^3} = 3x^2 y^4.$

Combine Both Sides:

Equating the simplified forms of both sides:

$4x^7 y^4 = 3x^2 y^4.$

Cancel Out Common Terms:

• The $y^4$ on both sides can be canceled (assuming $y \neq 0$):

[ 4x^7 =