Exponent Equation Simplification

The problem requires finding the specific value of the variable m that makes the following exponential equation true:

$$ \left(\frac{21}{8}\right)^{12} \times \left(\frac{8}{21}\right)^{14} \times \left(\frac{21}{8}\right)^{14} = \left(\frac{8}{21}\right)^{9m+20} $$

To solve this, we must simplify the equation by applying the fundamental rules of exponents, aiming to have the same base on both sides of the equation.

Applying Exponent Rules

Several key exponent rules are essential for simplifying this equation:

• Product of Powers: $a^x \times a^y = a^{x+y}$. This rule helps combine terms with the same base.
• Negative Exponent Rule: $\left(\frac{a}{b}\right)^{-x} = \left(\frac{b}{a}\right)^x$. This allows us to change the base to match other terms.
• Power of a Power Rule: $(a^x)^y = a^{xy}$. This rule is used when raising a power to another power.
• Equality Rule: If $a^x = a^y$ and the base $a$ is not $0, 1,$ or $-1$, then the exponents must be equal, i.e., $x = y$.

Our strategy is to express all parts of the equation using a single base, such as $\frac{21}{8}$. We can observe that $\left(\frac{8}{21}\right)$ is the reciprocal of $\left(\frac{21}{8}\right)$.

Step-by-Step Solution

Let's start with the given equation:

$$ \left(\frac{21}{8}\right)^{12} \times \left(\frac{8}{21}\right)^{14} \times \left(\frac{21}{8}\right)^{14} = \left(\frac{8}{21}\right)^{9m+20} $$

Using the negative exponent rule, we can rewrite $\left(\frac{8}{21}\right)$ as $\left(\frac{21}{8}\right)^{-1}$:

$$ \left(\frac{21}{8}\right)^{12} \times \left(\left(\frac{21}{8}\right)^{-1}\right)^{14} \times \left(\frac{21}{8}\right)^{14} = \left(\left(\frac{21}{8}\right)^{-1}\right)^{9m+20} $$

Apply the power of a power rule $(a^x)^y = a^{xy}$ to simplify the terms:

$$ \left(\frac{21}{8}\right)^{12} \times \left(\frac{21}{8}\right)^{(-1 \times 14)} \times \left(\frac{21}{8}\right)^{14} = \left(\frac{21}{8}\right)^{(-1 \times (9m+20))} $$

Simplify the exponents:

$$ \left(\frac{21}{8}\right)^{12} \times \left(\frac{21}{8}\right)^{-14} \times \left(\frac{21}{8}\right)^{14} = \left(\frac{21}{8}\right)^{-(9m+20)} $$

Now, combine the terms on the left side using the product of powers rule $a^x \times a^y = a^{x+y}$:

$$ \left(\frac{21}{8}\right)^{(12 + (-14) + 14)} = \left(\frac{21}{8}\right)^{-(9m+20)} $$

Calculate the sum of the exponents on the left side:

$$ \left(\frac{21}{8}\right)^{12} = \left(\frac{21}{8}\right)^{-(9m+20)} $$

With the bases now identical ($\frac{21}{8}$), we can equate the exponents based on the equality rule:

$$ 12 = -(9m+20) $$

Proceed to solve the linear equation for m:

$$ 12 = -9m - 20 $$

Add 20 to both sides of the equation:

$$ 12 + 20 = -9m $$

$$ 32 = -9m $$

Finally, divide both sides by -9 to isolate m:

$$ m = \frac{32}{-9} $$

$$ m = -\frac{32}{9} $$

Final Value of m

Therefore, the value of m that satisfies the given exponential equation is $m = -\frac{32}{9}$.