Let's simplify the expression step-by-step using the rules of exponents:
The given expression is:
[tex]\[ p^{\frac{1}{3}}\left( p^{\frac{2}{3}} - 4p^{\frac{8}{3}} \right) \][/tex]
First, distribute [tex]\( p^{\frac{1}{3}} \)[/tex] to each term inside the parentheses.
#### Step 1:
Distribute [tex]\( p^{\frac{1}{3}} \)[/tex] to [tex]\( p^{\frac{2}{3}} \)[/tex]:
[tex]\[ p^{\frac{1}{3}} \cdot p^{\frac{2}{3}} = p^{\frac{1}{3} + \frac{2}{3}} \][/tex]
Since the bases are the same, we add the exponents:
[tex]\[ \frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1 \][/tex]
So, this term simplifies to:
[tex]\[ p^1 = p \][/tex]
#### Step 2:
Distribute [tex]\( p^{\frac{1}{3}} \)[/tex] to [tex]\( -4p^{\frac{8}{3}} \)[/tex]:
[tex]\[ p^{\frac{1}{3}} \cdot (-4p^{\frac{8}{3}}) = -4p^{\frac{1}{3} + \frac{8}{3}} \][/tex]
Again, add the exponents:
[tex]\[ \frac{1}{3} + \frac{8}{3} = \frac{9}{3} = 3 \][/tex]
So, this term simplifies to:
[tex]\[ -4p^3 \][/tex]
#### Step 3:
Combine these simplified expressions:
[tex]\[ p - 4p^3 \][/tex]
Putting it all together, the simplified expression in exponential notation with positive exponents is:
[tex]\[ p - 4p^3 \][/tex]
