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Use the rules of exponents to simplify the expression. Write your answer with positive exponents. Assume that all variables represent positive real numbers.

[tex]\[
p^{\frac{1}{3}}\left(p^{\frac{2}{3}}-4 p^{\frac{8}{3}}\right)
\][/tex]

[tex]\[
p^{\frac{1}{3}}\left(p^{\frac{2}{3}}-4 p^{\frac{8}{3}}\right)=
\][/tex]

(Simplify your answer. Type exponential notation with positive exponents.)

Sagot :

GinnyAnsweridn
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]
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