Which law would you use to simplify the expression $\frac{3^{10}}{3^4}$?
A. quotient of powers
B. power of a quotient
C. product of powers
D. power of a product
Answer (1)
The problem requires identifying the correct exponent law to simplify 3 4 3 10 .
The quotient of powers law states that a n a m = a m − n .
The given expression matches the form of the quotient of powers.
Therefore, the quotient of powers law is the correct choice.
Explanation
Understanding the Problem We are asked to identify the law of exponents that simplifies the expression 3 4 3 10 . Let's analyze the given options.
Identifying Relevant Laws The expression 3 4 3 10 involves dividing two powers with the same base. The relevant exponent laws are:
Quotient of Powers: a n a m = a m − n
Power of a Quotient: ( b a ) n = b n a n
Product of Powers: a m ⋅ a n = a m + n
Power of a Product: ( ab ) n = a n b n
Applying the Correct Law The given expression 3 4 3 10 matches the form of the quotient of powers. Therefore, the quotient of powers law is the appropriate law to simplify the expression.
Conclusion The correct answer is quotient of powers.
Examples
The quotient of powers law is useful in various scientific and engineering applications. For example, when calculating the ratio of two light intensities or analyzing exponential decay processes, this law simplifies the calculations. Imagine you are comparing the brightness of two light sources, where one is 2 5 times brighter than a reference and the other is 2 2 times brighter. Using the quotient of powers, you can easily determine that the first light source is 2 2 2 5 = 2 5 − 2 = 2 3 = 8 times brighter than the second.
