Simplify (t^2)/(t^-3)
The given problem is asking to perform an algebraic simplification involving exponents. Specifically, it requires simplifying a rational expression where the numerator is 't' raised to the power of 2, denoted by \( t^2 \), and the denominator is 't' raised to the power of negative 3, denoted by \( t^{-3} \). The question involves applying the rules of exponents to combine the exponents on the variable 't' in the numerator and the denominator.
$\frac{t^{2}}{t^{- 3}}$
Answer
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:
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.
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 $n$, $a^m \cdot a^n = a^{m+n}$.
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.
