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The law of exponents is a fundamental concept in algebra that simplifies expressions involving powers of the same base. Imagine you have two numbers with the same base number and you want to divide or multiply them.
For division, the law of exponents tells you to subtract the exponents. This is written mathematically as \( \frac{x^a}{x^b} = x^{a-b} \). For multiplication, you would add the exponents, which looks like \( x^a \times x^b = x^{a+b} \).
In our exercise, we used this rule to simplify the variable parts of the monomial. By subtracting the exponent of the denominator from the exponent of the numerator, we obtain the simplified power, making calculations more straightforward. This powerful tool helps in reducing complexity in larger algebraic expressions.

Simplifying coefficients is the process of reducing the numerical part of a term in an algebraic expression. Coefficients are the numbers multiplied by the variables in a monomial. In this context, we simplify by division.
Let's take the example from the exercise: we have coefficients 65 and 5. By dividing these two numbers, we simplify 65/5 to become 13. The division makes it easier to manage and interprets your algebraic expression.
Simplifying coefficients right away can help in visualizing the larger picture of the problem. It reduces the chance of error when you deal with more complex equations or multi-step problems.

Navigating variable powers in algebra is key to mastering expressions. Algebra often deals with variables raised to powers, like \( x^2 \) or \( y^3 \). When dividing monomials, dealing with these powers effectively is crucial.
To simplify a term involving variable powers, apply the law of exponents as mentioned before. This involves subtracting the lower exponent from the higher one. For example, in \( \frac{x^2}{x^1} \), we subtract 1 from 2, resulting in \( x^{2-1} = x \).
Understanding and managing variable powers allows you to simplify complex algebraic expressions efficiently. By breaking down these terms, you gain a clearer understanding of the equation, making further calculation or adjustment simpler.