The Quotient Rule: 동영상 및 연습문제

When dividing exponential expressions with the same base, the quotient rule of exponents allows us to simplify by subtracting the exponents. For example, dividing \$4^3$ by \$4^1$ can be understood by expanding both terms: \$4^3$ is \(4 \times 4 \times 4\), and \$4^1$ is simply \(4\). Canceling one \(4\) from the numerator and denominator leaves \(4 \times 4\), which is \$4^2$. This matches the result of subtracting the exponents: \(3 - 1 = 2\). Thus, the quotient rule states that for any nonzero base $a$,

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

where $m$ and $n$ are integers, and \(a \neq 0\). This rule is intuitive when considering the division and subtraction symbols both involve a bar, linking the operation of division with subtracting exponents. Conversely, when multiplying exponential expressions with the same base, the exponents are added.

Applying this rule to variables, if we have $y^7$ divided by $y^5$, the expression simplifies to $y^{7-5} = y^2$. Similarly, dividing $m^6$ by $m^6$ results in $m^{6-6} = m^0$. This introduces the important zero exponent rule, which states that any nonzero number raised to the zero power equals one:

\[a^0 = 1 \quad \text{for} \quad a \neq 0\]

This can be verified by expanding both numerator and denominator and canceling identical factors, leaving the value 1. Understanding these exponent rules is essential for simplifying expressions efficiently and forms a foundation for more advanced algebraic manipulations.

To simplify the expression nine sixteenths divided by three fourths, it is helpful to rewrite the numbers using exponential expressions with the same base. Recognizing that 9 can be expressed as 3 squared (\$3^2$) and 16 as 4 squared (\$4^2$), the fraction \(\frac{9}{16}\) becomes \(\frac{3^2}{4^2}\). The divisor, \(\frac{3}{4}\), remains as is.

Applying the quotient rule for exponents, which states that when dividing like bases you subtract the exponents (\(\frac{a^m}{a^n} = a^{m-n}\)), we can rewrite the original expression as:

\[ \frac{\frac{3^2}{4^2}}{\frac{3}{4}} = \frac{3^2}{3} \div \frac{4^2}{4} \]

For the numerator, dividing \$3^2$ by \(3\) is equivalent to \(3^{2-1} = 3^1 = 3\). Similarly, for the denominator, dividing \$4^2$ by \(4\) gives \(4^{2-1} = 4^1 = 4\). Thus, the simplified expression becomes:

\[ \frac{3}{4} \]>

This demonstrates how the quotient rule for exponents can be used to simplify complex fractional expressions by rewriting terms with common bases and subtracting exponents accordingly.