Question 201·Hard·Equivalent Expressions

Once you have simplified $(27 m^{-4/3} n^{1/2})^{2/3}$, multiply it by $m^{1/2} n^{-5/4}$. Remember: when you multiply powers with the same base, you add the exponents.

Write the whole expression as a single fraction. When dividing by $n^{-1/6}$, think carefully about subtracting a negative exponent. After simplifying, convert any negative exponents into positive ones by moving those factors to the denominator.

Type the original expression into Desmos exactly as given, for example as (27*m^(-4/3)*n^(1/2))^(2/3)*m^(1/2)*n^(-5/4)/(3*n^(-1/6)). Desmos will create sliders for $m$ and $n$; set them to positive values (like $m=4$, $n=9$) since $m>0$ and $n>0$.

On new lines, type each answer choice expression (A through D) using the same $m$ and $n$ variables. For example, one line for 3/(m^(7/9)*n^(3/4)), another for 3/(m^(7/18)*n^(5/12)), another for 3*m^(7/18)*n^(3/4), and another for 3/(m^(7/18)*n^(3/4)).

Change the slider values of $m$ and $n$ to several different positive values (for example, $m=1$, $n=2$; then $m=5$, $n=3$). For each pair $(m,n)$, compare the numeric value of the original expression to each answer choice. The correct answer is the one whose value always matches the original expression for all tested positive values of $m$ and $n$.

Start with the part

Use $(ab)^k = a^k b^k$ and $(x^a)^b = x^{ab}$:

• $27^{2/3}$: since $27^{1/3} = 3$, then $27^{2/3} = (27^{1/3})^2 = 3^2 = 9$.
• $m^{-4/3}$ raised to $2/3$ gives $m^{(-4/3)(2/3)} = m^{-8/9}$.
• $n^{1/2}$ raised to $2/3$ gives $n^{(1/2)(2/3)} = n^{1/3}$.

So

Now the whole numerator is

Combine like bases in

For like bases, add exponents:

• For $m$: $-8/9 + 1/2$.

  • Common denominator $18$: $-8/9 = -16/18$ and $1/2 = 9/18$.
  • Sum: $-16/18 + 9/18 = -7/18$, so we get $m^{-7/18}$.

• For $n$: $1/3 + (-5/4)$.

  • Common denominator $12$: $1/3 = 4/12$ and $-5/4 = -15/12$.
  • Sum: $4/12 - 15/12 = -11/12$, so we get $n^{-11/12}$.

So the numerator simplifies to

The whole expression is now

Now simplify

First, simplify the numerical coefficients:

• $9/3 = 3$.

Next, handle each variable:

• There is no $m$ in the denominator, so the exponent on $m$ stays $-7/18$.
• For $n$, when dividing powers with the same base, subtract exponents:
  • Exponent in numerator: $-11/12$.
  • Exponent in denominator: $-1/6$.
  • New exponent: $-11/12 - (-1/6) = -11/12 + 1/6$.
  • Convert $1/6$ to twelfths: $1/6 = 2/12$, so

• So we get $n^{-3/4}$.

The expression is now

Use the rule $x^{-a} = 1/x^a$ to move factors with negative exponents to the denominator:

This matches answer choice D, so the expression is equivalent to