Which is equivalent to $\frac{b^a}{b^m}$ ?
To determine the equivalent expression for \(\frac{b^a}{b^m}\), we apply the quotient rule for exponents. This fundamental law states that when dividing two powers with the same base, you subtract the exponent of the divisor from the exponent of the dividend. Formally, for any non-zero real number \(b\) and exponents \(a\) and \(m\), the rule is defined as:
\[ \frac{b^a}{b^m} = b^{a-m} \]
This rule arises because \(b^a = \underbrace{b \times b \times \dots \times b}_{a \text{ factors}}\) and \(b^m = \underbrace{b \times b \times \dots \times b}_{m \text{ factors}}\). Dividing \(b^a\) by \(b^m\) cancels out \(m\) factors of \(b\) from the numerator, leaving \(a - m\) factors of \(b\).
Comparing our derived expression, \(b^{a-m}\), with the provided options:
It is clear that the first option, \(b^{a-m}\), is the correct equivalent. Therefore, the equivalent expression to \(\frac{b^a}{b^m}\) is \(\boxed{b^{a-m}}\).
Exponent laws are essential in various fields, such as computer science, engineering, and finance. For instance, in data storage, if you need to know how many times larger a terabyte is compared to a gigabyte, you can use the quotient rule: \(\frac{10^{12}}{10^9} = 10^{12-9} = 10^3 = 1000\). This demonstrates how exponent rules simplify calculations involving large numbers and help in understanding scales and proportions efficiently.