[Math] Squaring both sides when units are different

algebra-precalculusarithmeticnumber theory

Given $((9) \text{inches})^{1/2} = ((0.25) \text{yards})^{1/2}$, then which of the following statements is true?

  1. $((3) \text{inches}) = ((0.5) \text{yards})$
  2. $((9) \text{inches}) = ((1.5) \text{yards})$
  3. $((9) \text{inches}) = ((0.25) \text{yards})$
  4. $((81) \text{inches}) = ((0.0625) \text{yards})$

My question is : Can I apply here as $x^{1/2}=y^{1/2}$ then square both sides $\implies x=y$. But as given units are different. So,

Can you explain it, please?

Best Answer

If (one thing)$^\frac12$ = (another thing)$^\frac12$, then we can square both sides to get

one thing = another thing

Here, "one thing" is "$9$ inches", and "another thing" is "$0.25$ yards" (I don't know why you added all those parentheses!). Hence

$9$ inches = $0.25$ yards

Your worry about the units is irrelevant here, because they are inside the outer parentheses. This question is a reminder that $(9$ inches$)^\frac12$ is not the same as $9^\frac12$ inches.

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