Question 112·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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The positive numbers $k$ , $m$ , and $n$ satisfy

k = 5m^2 n

Which equation correctly expresses $m$ in terms of $k$ and $n$ ?

The phrase “express $m$ in terms of $k$ and $n$ ” signals rearranging a formula: get $m$ alone while leaving the other letters in the answer. Undo the operations around $m$ in reverse order, dividing away its multiplying factor before taking a square root. Squaring hides whether a number was positive or negative, so use the given positivity condition to choose the sign.

Hints

Hint 1
$m^2$ is multiplied by $5n$ , so divide both sides by $5n$ to undo that multiplication. Since $n$ is positive, $5n$ cannot be zero. What does the resulting fraction equal?
Hint 2
A square root undoes a square, but positive and negative numbers can have the same square. Once $m^2$ is alone, take the root of the whole fraction. Which sign is allowed for $m$ ?

Step-by-step

Undo the multiplication, then the square
Step 1
Test the equation with positive values
Choose the positive sample $k=20$ and $n=1$ to test the given equation. Type $k=20$ , $n=1$ , and $k\sim5m_1^2n\{m_1>0\}$ . Desmos uses $m_1$ for the value to find and reports $m_1=2$ under PARAMETERS. The restriction keeps the positive solution. A sample checks a formula, but it cannot prove the formula works for every positive pair.
Step 2
Get the square of $m$ alone
A formula in $k$ and $n$ must work for every positive pair, so rearrange the letters by hand.

To get $m$ alone, isolate $m^2$ before taking a root. Since $5n$ multiplies $m^2$ and $n>0$ , divide both sides by $5n$ :
$\frac{k}{5n}=m^2.$
Step 3
Undo the square
Taking a square root undoes squaring, but both signs are possible because a number and its negative have the same square. Take the root of the whole fraction:
$m=\pm\sqrt{\frac{k}{5n}}.$
Step 4
Choose the positive value
The problem says $m$ is positive, so the negative possibility cannot be $m$ . Keep the positive one:
$m=\sqrt{\frac{k}{5n}}.$
To check it against the sample, type $\sqrt{\frac{k}{5n}}$ . Desmos prints $2$ , matching the value it found for $m_1$ .

This expresses $m$ in terms of $k$ and $n$ . Choice D.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 112·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this