91Ó°ÊÓ

The quotient rule of exponents is used to simplify expressions involving the division of two exponential terms with the same base. This rule states that when you divide powers with the same base, you subtract the exponents.

For example, \[ \frac{a^m}{a^n} = a^{m-n} \]

In our exercise, we used this rule on the expression \[ \frac{n^{-15}}{n^{5}} \]. Here, both terms have the base 'n'. Applying the quotient rule, we subtract the exponent in the denominator from the exponent in the numerator:

\[ n^{-15-5}=n^{-20} \]

So, \[ \frac{n^{-15}}{n^{5}} \] simplifies to \[ n^{-20} \]. This makes complex expressions more manageable by reducing the number of terms.

Negative exponents can be tricky, but they follow a straightforward rule: a negative exponent indicates that the base should be taken as the reciprocal.

For example, \[ a^{-m} = \frac{1}{a^m} \]

In our exercise, we ended up with \[ 2 \times n^{-20} \]. This can be rewritten using the negative exponent rule:

\[ 2 \times \frac{1}{n^{20}} = \frac{2}{n^{20}} \]

By understanding negative exponents, we can transform expressions into different but equivalent forms, making them easier to interpret and use.

Algebraic simplification involves reducing an expression to its simplest form by following a series of steps and rules. In our example, we simplified \[ \frac{10 n^{-15}}{5 n^{5}} \] using these key steps:

1. **Simplify the Constants**: Divide the numerical coefficients (constants) first: \[ \frac{10}{5} = 2 \]
2. **Apply the Quotient Rule**: Simplify exponents by subtracting the exponents in the denominator from those in the numerator: \[ \frac{n^{-15}}{n^{5}} = n^{-20} \]
3. **Combine Results**: Multiply the simplified constants by the simplified variable term: \[ 2 \times n^{-20} = 2n^{-20} \]

Understanding and applying these simplification techniques makes tackling complex algebraic problems much easier.