Simplify (t^2)/(t^-3)

Solution:

Simplification Process

Step 1:

Apply the rule for negative exponents to move $t^{-3}$ from the denominator to the numerator: $\frac{1}{t^{-3}}=t^3$. Hence, the expression becomes $t^2\cdot t^3$.

Step 2:

Combine the terms with the same base by summing their exponents.

Step 2.1:

Invoke the exponentiation rule which states $a^m\cdot a^n=a^{m+n}$. Thus, we have $t^{2+3}$.

Step 2.2:

Perform the addition of the exponents: $2+3=5$. The expression simplifies to $t^5$.

Knowledge Notes:

To solve the given problem, a few algebraic rules regarding exponents are applied:

1. Negative Exponent Rule: For any non-zero number $a$ and any integer $n$, $a^{-n}=\frac{1}{a^n}$. This rule allows us to transform a negative exponent into a positive one by moving the base from the denominator to the numerator or vice versa.

2. Power Rule: When multiplying two expressions with the same base, you can add the exponents. Formally, for any real number $a$ (except 0) and integers $m$ and $n$, $a^m\cdot a^n=a^{m+n}$.

3. Simplification: The process of simplification often involves applying these rules to make an expression easier to understand or work with, often by reducing it to a more compact form.

In the context of the given problem, these rules are used to simplify the expression $(t^2)/(t^{-3})$ to $t^5$. The negative exponent is first converted to a positive exponent, which allows for the multiplication of terms with the same base. The exponents are then added together to arrive at the final simplified form.