Write:
[tex]a^6 \cdot a^5 \div a^2[/tex] as a power to the base [tex]a^3[/tex]
Answer (1)
Simplify the expression using exponent rules: a 6 "." a 5 ÷ a 2 = a 6 + 5 − 2 = a 9 .
Express a 9 as a power of a 3 : a 9 = ( a 3 ) k .
Equate the exponents: 9 = 3 k , which gives k = 3 .
Therefore, a 6 "." a 5 ÷ a 2 = ( a 3 ) 3 , and the answer is ( a 3 ) 3 .
Explanation
Understanding the Problem We are asked to express a 6 "." a 5 ÷ a 2 as a power of a 3 . This means we want to find an exponent k such that a 6 "." a 5 ÷ a 2 = ( a 3 ) k .
Simplifying the Expression First, let's simplify the left-hand side of the equation using the properties of exponents. Recall that when multiplying powers with the same base, we add the exponents: a m "." a n = a m + n . Also, when dividing powers with the same base, we subtract the exponents: a m ÷ a n = a m − n .
Calculating the Exponent Applying these rules, we have:
a 6 "." a 5 ÷ a 2 = a 6 + 5 ÷ a 2 = a 11 ÷ a 2 = a 11 − 2 = a 9 .
Expressing as a Power of a^3 Now we want to express a 9 as a power of a 3 . That is, we want to find k such that a 9 = ( a 3 ) k . Recall that when raising a power to a power, we multiply the exponents: ( a m ) n = a m ⋅ n . So, ( a 3 ) k = a 3 k .
Solving for k Therefore, we have a 9 = a 3 k . Since the bases are equal, we can equate the exponents: 9 = 3 k . Solving for k , we get k = 3 9 = 3 .
Final Answer Thus, a 6 "." a 5 ÷ a 2 = a 9 = ( a 3 ) 3 . So, the expression a 6 "." a 5 ÷ a 2 as a power to the base a 3 is ( a 3 ) 3 .
Examples
Understanding how to manipulate exponents is crucial in many scientific fields. For example, in computer science, memory sizes are often expressed as powers of 2 (e.g., kilobytes, megabytes, gigabytes). Simplifying expressions with exponents helps in calculating storage requirements and data transfer rates. Similarly, in physics, understanding exponential relationships is essential in fields like radioactive decay and wave mechanics, where quantities change exponentially over time or distance.
