Verify the mathematical correctness of an algebraic identity involving absolute value and fraction manipulation | Step-by-Step Solution

Let's Explore This Algebraic Identity Together! 🔍

1. What We're Solving:

We need to determine whether the identity $\frac{|a|}{a^2} = \frac{1}{|a|}$ is always true for all valid values of $a$. This means checking if these two expressions are mathematically equivalent.

2. The Approach:

To verify an algebraic identity, we have a few strategies:

• Direct algebraic manipulation: Transform one side to match the other
• Test with specific values: See if both sides give the same results
• Consider domain restrictions: Make sure both sides are defined for the same values

We'll use algebraic manipulation as our main approach.

3. Step-by-Step Solution:

Step 1: Identify the domain Both expressions involve division, so we need $a \neq 0$ (can't divide by zero!). Also, the right side has $|a|$ in the denominator, which reinforces that $a \neq 0$.

Step 2: Recall key properties Remember that for any non-zero number $a$:

• $a^2 = |a|^2$ (squaring always gives a positive result)
• $|a| > 0$ when $a \neq 0$

Step 3: Simplify the left side Let's work with $\frac{|a|}{a^2}$:

Since $a^2 = |a|^2$, we can rewrite this as: $$\frac{|a|}{a^2} = \frac{|a|}{|a|^2}$$

Step 4: Apply fraction rules Using the rule $\frac{x}{x^2} = \frac{1}{x}$ (when $x \neq 0$): $$\frac{|a|}{|a|^2} = \frac{1}{|a|}$$

Step 5: Compare both sides Left side after simplification: $\frac{1}{|a|}$ Right side: $\frac{1}{|a|}$

They match! ✨

4. The Answer:

YES, the identity is correct!

The identity $\frac{|a|}{a^2} = \frac{1}{|a|}$ is true for all $a \neq 0$.

We proved this by recognizing that $a^2 = |a|^2$, which allowed us to simplify the left side to exactly match the right side.

5. Memory Tip:

Think of it this way: "When I square any number, I get the same result as squaring its absolute value!" So $a^2$ and $|a|^2$ are always the same. This makes the fraction $\frac{|a|}{a^2}$ become $\frac{|a|}{|a|^2}$, which simplifies just like $\frac{3}{3^2} = \frac{3}{9} = \frac{1}{3}$!

Great job working through this identity - you're building strong algebraic reasoning skills! 🎉