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Exponents are powerful tools that help us simplify expressions. One key property is the **quotient of powers property**. This property states that for any non-zero base \(a\), and integers \(m\) and \(n\), the following holds true: \ \( \frac{a^m}{a^n} = a^{m-n} \). This property allows us to subtract exponents when dividing like bases. For example, in the exercise \( \frac{3^{8} y^{5}}{3^{10} y^{2}} \), we applied this property twice: \

• First, to the numerical bases: \( \frac{3^{8}}{3^{10}} = 3^{8-10} = 3^{-2} \)
• Second, to the variable bases: \( \frac{y^{5}}{y^{2}} = y^{5-2} = y^{3} \)

Understanding these properties helps us to manage and simplify expressions efficiently.

When working with exponents, it's important to understand how to handle numerical bases. A numerical base is simply a number raised to an exponent. In our exercise, we had the numerical base \(3\). By applying the quotient of powers property, we simplified \( \frac{3^{8}}{3^{10}} \) to \( 3^{-2} \). \ Why do we subtract exponents here? It's because we are dividing powers with the same base. Here’s a quick reminder:

• \( 3^{8} \) represents \(3\) multiplied by itself 8 times.
• \( 3^{10} \) represents \(3\) multiplied by itself 10 times.

When you divide \(3^{8}\) by \(3^{10}\), you cancel out 8 factors of \(3\), leaving you with \( 3^{-2} \), or \( \frac{1}{3^2} \), which is \( \frac{1}{9} \). Understanding this makes it easier to see why expressions like \( \frac{3^{8}}{3^{10}} \) simplify as they do.

Variable bases work similarly to numerical bases when it comes to exponents. A variable base is usually a letter that represents a number, like \( y \) in our exercise. Using the same properties of exponents, we find that: \ \( \frac{y^{5}}{y^{2}} = y^{5-2} = y^{3} \). Let's break this down: \

• \( y^{5} \) means \( y \) multiplied by itself 5 times.
• \( y^{2} \) means \( y \) multiplied by itself 2 times.

When dividing these, the 2 \( y \)s in the denominator cancel out 2 of the 5 \( y \)s in the numerator, leaving \( y^3 \). Simplifying variable bases follows the same intuitive rules as numerical bases, making them easier to manage.