Simplify (p^8)/(p^3)
The question is asking for the simplification of a mathematical expression involving exponents. Specifically, you are being asked to divide two powers of the same base 'p'. Here, you have p raised to the 8th power (p^8) as the numerator and p raised to the 3rd power (p^3) as the denominator. The problem demands applying the laws of exponents to reduce the expression to its simplest form.
$\frac{p^{8}}{p^{3}}$
Answer
Solution:
Simplification Process:
Step 1: Break down $p^{8}$ by separating $p^{3}$ from it. Write it as $\frac{p^{3} \cdot p^{5}}{p^{3}}$.
Step 2: Eliminate identical factors from numerator and denominator.
Step 2.1: Introduce multiplication by $1$ to clarify the process: $\frac{p^{3} \cdot p^{5}}{p^{3} \cdot 1}$.
Step 2.2: Strike out the matching factors: $\frac{\cancel{p^{3}} \cdot p^{5}}{\cancel{p^{3}} \cdot 1}$.
Step 2.3: Reformulate the fraction: $\frac{p^{5}}{1}$.
Step 2.4: Compute the division of $p^{5}$ by $1$: $p^{5}$.
Knowledge Notes:
When simplifying expressions with exponents, we use the laws of exponents. Here are the relevant points:
Multiplication of Like Bases: When multiplying powers with the same base, you add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
Division of Like Bases: When dividing powers with the same base, you subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$, provided $a \neq 0$.
Exponent of One: Any number raised to the power of one is the number itself, e.g., $a^1 = a$.
Exponent of Zero: Any non-zero number raised to the power of zero is one, e.g., $a^0 = 1$.
Cancellation Law: When a factor appears in both the numerator and the denominator of a fraction, it can be canceled out. This is because any number divided by itself equals one.
In the given problem, we apply the division law of exponents to simplify the expression $\frac{p^8}{p^3}$ by subtracting the exponents. We get $p^{8-3} = p^5$. Since dividing by 1 does not change the value of a number, $\frac{p^5}{1}$ simplifies to $p^5$.
