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Exponents play a crucial role in algebra as they are used to express repeated multiplication in a concise form. When we see a term like \( t^5 \), the exponent is the 5, indicating that the base \( t \) is multiplied by itself a total of 5 times: \( t \times t \times t \times t \times t \).
This simplification is what makes working with large numbers and complex equations manageable.

• When the bases are the same, we can add the exponents during multiplication.
• This applies to any number or variable raised to an exponent.

For example, in the problem, the numerator has terms \( t^5 \), \( t^6 \), and \( t \), all with the same base. Combining these by adding the exponents gives:

• \( t^5 \times t^6 \times t = t^{5+6+1} = t^{12} \).

Simplifying algebraic expressions involves reducing them to their simplest form. This often involves following a series of mathematical rules and properties. In the exercise, we simplify the expression by:

• Combining like terms.
• Reducing fractions by canceling out common terms in the numerator and the denominator.

The given expression \( \frac{t^{5} t^{6} t}{t^{2} t^{3}} \) requires us first to simplify the terms by combining and then reducing them using the properties of exponents and algebraic simplification. To accomplish this:

• First combine the terms in the numerator and denominator separately, as seen in the solution steps.
• Simplifying further by using the division of exponents property later.

Simplification is not just about doing less work but getting to a form that is easier to handle, both for interpretation and further calculations.

Division involving exponents follows a specific rule where you subtract the exponents of like bases. This operation is based on the principle that dividing repeated multiplication results in the terms cancelling each other out. This rule can be written as:\[ \frac{a^m}{a^n} = a^{m-n} \]where \( a \) is the base and \( m \), \( n \) are the exponents.
In our problem, after simplifying the numerator and denominator, we had exactly these conditions:

• \( \frac{t^{12}}{t^5} \).

Applying our division of exponents rule here:

• Subtract the exponent in the denominator from the exponent in the numerator: \( t^{12-5} \), which results in \( t^7 \). This is the simplest form we could get for this expression, adhering strictly to the exponent rules.

The result \( t^7 \) indicates that in the division step, the common terms from the repeated multiplication in the numerator and denominator have effectively been cancelled out by subtraction of their exponents, revealing the base raised to the final exponent.