Sure! Let's solve the equation step-by-step.
We are given:
[tex]\[
\left( \frac{1}{2} \right)^{-5p} \div \left( \frac{1}{2} \right)^{3p} = \left( \frac{1}{2} \right)^{24}
\][/tex]
Step 1: Use properties of exponents to simplify the left side. Recall that dividing exponents with the same base can be converted into subtraction of exponents:
[tex]\[
\left( \frac{1}{2} \right)^{-5p} \div \left( \frac{1}{2} \right)^{3p} = \left( \frac{1}{2} \right)^{-5p - 3p}
\][/tex]
Simplifying the exponent on the left side:
[tex]\[
\left( \frac{1}{2} \right)^{-5p - 3p} = \left( \frac{1}{2} \right)^{-8p}
\][/tex]
Step 2: Now, we have:
[tex]\[
\left( \frac{1}{2} \right)^{-8p} = \left( \frac{1}{2} \right)^{24}
\][/tex]
Step 3: Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
-8p = 24
\][/tex]
Step 4: Solve for [tex]\( p \)[/tex]. To isolate [tex]\( p \)[/tex], divide both sides of the equation by [tex]\(-8\)[/tex]:
[tex]\[
p = \frac{24}{-8} = -3
\][/tex]
Therefore, the value of [tex]\( p \)[/tex] is:
[tex]\[
p = -3.0
\][/tex]
