The positive numbers , , and satisfy the equation Which equation correctly expresses in terms of and ?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 88 | Free SAT Prep | Aniko

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Question 88·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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

mn^2 = 8k

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

For this type of SAT question, treat it like solving an equation for a variable, even though the other symbols are letters instead of numbers. First isolate the term containing that variable (here, divide both sides by the coefficient of $n^2$ ), then undo exponents by using roots, keeping track of when you must choose the positive root because the variable is given as positive. Finally, compare your simplified expression exactly to the answer choices, paying close attention to whether variables are in the numerator or denominator and whether they are inside or outside square roots.

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Get $n^2$ alone first

Look at the equation $mn^2 = 8k$ . What operation will remove the $m$ from the left side so that only $n^2$ is left?

Deal with the exponent on $n$

Once you have an equation of the form $n^2 = \text{(something)}$ , what operation will get you from $n^2$ to $n$ ?

Use the fact that the variables are positive

After taking a square root, remember that $n$ is given to be positive. How does that affect whether you take the positive or negative root?

Desmos Guide

Set up the original equation

In Desmos, type m*n^2 = 8*k so you can refer back to this relationship. Treat $m$ and $k$ as positive constants (you can add sliders for them if you like).

Test each expression for $n$

For each answer choice, define $n$ using that expression (for example, n1 = (8*k)/(m^2), n2 = sqrt(8*k*m), etc.). Then, for each defined $n$ expression, form the left-hand side m*(ni)^2 and compare it to 8*k.

Check which expression always satisfies the equation

Adjust the sliders for $m$ and $k$ to different positive values. The correct formula for $n$ will make m*(ni)^2 equal to 8*k for all positive $m$ and $k$ , while the incorrect ones will not match 8*k consistently.

Step-by-step Explanation

Isolate the term with $n$

You are given

mn^2 = 8k.

To get $n$ by itself, first isolate $n^2$ by dividing both sides of the equation by $m$ (since $m$ is positive, dividing by $m$ is allowed):

\frac{mn^2}{m} = \frac{8k}{m}

which simplifies to

n^2 = \frac{8k}{m}.

Undo the square on $n$

Now you have

n^2 = \frac{8k}{m}.

To solve for $n$ , take the square root of both sides:

\sqrt{n^2} = \sqrt{\frac{8k}{m}}.

The left side becomes $|n|$ . So

|n| = \sqrt{\frac{8k}{m}}.

Use the fact that $n$ is positive and match the choice

Because the problem states that $n$ is a positive number, $|n|$ equals $n$ (not $-n$ ). So

n = \sqrt{\frac{8k}{m}}.

Now match this with the answer choices: this expression is exactly choice C.

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Step-by-step Explanation

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Step-by-step Explanation