Apply the product rule of integer exponents to find the numerical equivalent of 12^−5⋅12^7 . Show your work and explain how you used the product rule of integer exponents.(2 points)

Answer 1

To solve the expression ${12}^{-5}\cdot {12}^7$, we will apply the product rule of integer exponents, which states that when multiplying two expressions with the same base, you can add their exponents. The rule can be written as:

$$a^m\cdot a^n=a^{m+n}$$

In this case, our base $a$ is $12$, $m$ is $-5$, and $n$ is $7$.

Using the product rule, we can combine the exponents:

$${12}^{-5}\cdot {12}^7={12}^{-5+7}$$

Now, we perform the addition of the exponents:

$$-5+7=2$$

Thus, we have:

$${12}^{-5}\cdot {12}^7={12}^2$$

Next, we can calculate ${12}^2$:

$${12}^2=12\cdot 12=144$$

So, the numerical equivalent of ${12}^{-5}\cdot {12}^7$ is:

$$\boxed{{\displaystyle 144}}$$

In summary, we applied the product rule of integer exponents to combine the exponents of the same base and calculated the resulting exponent to find the final numerical value.