Solving for m: Exponential Equation with Fractions

The problem requires finding the value of m in the given exponential equation:

$ \left(\frac{28}{9}\right)^{11} \times \left(\frac{9}{28}\right)^{16} \times \left(\frac{28}{9}\right)^{17} = \left(\frac{9}{28}\right)^{2m+8} $

Simplify the Left Side

To solve the equation, we first simplify the left-hand side (LHS) by expressing all terms with the same base, preferably $\left(\frac{28}{9}\right)$. We use the property $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$.

• Rewrite $\left(\frac{9}{28}\right)^{16}$ as $\left(\frac{28}{9}\right)^{-16}$.
• The equation becomes: $ \left(\frac{28}{9}\right)^{11} \times \left(\frac{28}{9}\right)^{-16} \times \left(\frac{28}{9}\right)^{17} $
• Combine the exponents using the rule $a^x \times a^y \times a^z = a^{x+y+z}$:
$ \left(\frac{28}{9}\right)^{11 + (-16) + 17} $ $ \left(\frac{28}{9}\right)^{11 - 16 + 17} $ $ \left(\frac{28}{9}\right)^{12} $

So, the simplified LHS is $\left(\frac{28}{9}\right)^{12}$.

Simplify the Right Side

Now, simplify the right-hand side (RHS) by expressing it with the base $\left(\frac{28}{9}\right)$.

• Using the property $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$, we rewrite $\left(\frac{9}{28}\right)^{2m+8}$ as:
$ \left(\frac{28}{9}\right)^{-(2m+8)} $ $ \left(\frac{28}{9}\right)^{-2m-8} $

Equate Exponents and Solve for m

The equation now is:

$ \left(\frac{28}{9}\right)^{12} = \left(\frac{28}{9}\right)^{-2m-8} $

Since the bases are the same ($\left(\frac{28}{9}\right)$), the exponents must be equal:

$ 12 = -2m - 8 $

Now, solve for m:

• Add 8 to both sides:
$ 12 + 8 = -2m $ $ 20 = -2m $
• Divide by -2:
$ m = \frac{20}{-2} $ $ m = -10 $

The value of m is -10.