Simplify the expression \(\left( \frac{2x}{8x^2} \right)^{-2}\).
Understand the Problem
The question asks to simplify the expression (\left( \frac{2x}{8x^2} \right)^{-2}). This involves understanding exponent rules, fraction manipulation, and simplification of algebraic expressions.
Answer
$16x^2$
Answer for screen readers
$16x^2$
Steps to Solve
- Simplify the fraction inside the parentheses
Start by simplifying the fraction $\frac{2x}{8x^2}$ by dividing both the numerator and the denominator by their common factors. The greatest common factor of $2x$ and $8x^2$ is $2x$. $$ \frac{2x}{8x^2} = \frac{2x \div 2x}{8x^2 \div 2x} = \frac{1}{4x} $$
- Apply the negative exponent
Now apply the exponent $-2$ to the simplified fraction $\frac{1}{4x}$. Recall that a negative exponent means taking the reciprocal of the base raised to the positive exponent. Therefore, $$ \left( \frac{1}{4x} \right)^{-2} = \left( \frac{4x}{1} \right)^{2} = (4x)^2 $$
- Apply the exponent to the term
Finally, apply the exponent $2$ to both the constant and the variable inside the parentheses. $$ (4x)^2 = 4^2 \cdot x^2 = 16x^2 $$
$16x^2$
More Information
The expression simplifies to $16x^2$. This is a quadratic term, meaning it represents a parabolic relationship if graphed.
Tips
A common mistake is not properly applying the negative exponent which leads to not inverting the fraction. Another mistake is only applying the exponent to $x$ and not to the coefficient 4.
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