Question

How do you find $p\left( -1 \right)$ given $p\left( a \right)=-{{4}^{3a}}$?

Answer 1

Hint: We have been a function $p$ in a-variable. This function is an exponential function whose base is equal to -4 and the exponent is expressed as the function of variable-a. We have to find the value of $p\left( -1 \right)$which we will calculate by substituting $-1$ in place of $a$ in the given function. Further, we shall open the brackets and multiply each term by 3 to perform the required multiplication and simplification of the terms.

Complete step by step solution:
Given that $p\left( a \right)=-{{4}^{3a}}$.
This is an exponential function with a negative base and the power as a function of variable-a.
Now, the expression $p\left( -1 \right)$ represents the same function$p$in constant value, $\left( -1 \right)$ instead of the variable-a. Another way of understanding it would be that the given function is simply written in terms of $\left( -1 \right)$ instead of variable-a.
Thus, in order to find $p\left( -1 \right)$, we shall substitute variable-a with $\left( -1 \right)$ at all places where variable-a is written in the function.
$\Rightarrow p\left( a \right)=-{{4}^{3a}}$
Substituting variable-a with $\left( -1 \right)$, we get
$\Rightarrow p\left( -1 \right)=-{{4}^{3\left( -1 \right)}}$
We shall now open the brackets of $\left( -1 \right)$ and multiply it by 3.
$\Rightarrow p\left( -1 \right)=-{{4}^{-3}}$
From the properties of exponential functions, we know that ${{a}^{-b}}=\dfrac{1}{{{a}^{b}}}$ where $a$ is the base of the exponential expression and $b$ is the power of the exponential expression.
$\Rightarrow p\left( -1 \right)=-\dfrac{1}{{{4}^{3}}}$
Here, ${{4}^{3}}$ represents multiplying 4 by itself 3 times.
$\Rightarrow p\left( -1 \right)=-\dfrac{1}{64}$
Therefore, if $p\left( a \right)=-{{4}^{3a}}$, then $p\left( -1 \right)$ is equal to $-\dfrac{1}{64}$.

Note:
One possible mistake we could have done while solving this problem would be while opening the brackets of the formed new expression. Sometimes, we tend to ignore the negative sign used between the terms within such brackets and write them as positive terms and thus, we could have written 3 instead of -3 in the power of the exponential expression.