Simplify for real m≠0.

01 / Standard#Your turn

\sqrt{1+\left(\frac{m^4-1}{2m^2}\right)^2}

Official paper · jm01-2024 · I.9 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Put the expression inside the root over a common denominator.

Hint 2

Check the sign before removing the square root.

Worked solution

The numerator is a perfect square.
$1+\frac{(m^4-1)^2}{4m^4}=\frac{(m^4+1)^2}{4m^4}$
Both m⁴+1 and 2m² are positive.
$\sqrt{\frac{(m^4+1)^2}{4m^4}}=\frac{m^4+1}{2m^2}=\frac{m^2}2+\frac1{2m^2}$

Checks and common pitfalls: Square roots return the nonnegative value; m=0 is excluded.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The numerator is a perfect square.
$1+\frac{(m^4-1)^2}{4m^4}=\frac{(m^4+1)^2}{4m^4}$
Both m⁴+1 and 2m² are positive.
$\sqrt{\frac{(m^4+1)^2}{4m^4}}=\frac{m^4+1}{2m^2}=\frac{m^2}2+\frac1{2m^2}$

Think first. Reveal a hint when the class is ready.

2024 JM01