Write the expression as an exponent with the base of 3: [tex]$81 \cdot 3^6$[/tex]
Answer (1)
Rewrite 81 as a power of 3: 81 = 3 4 .
Substitute 3 4 into the original expression: 3 4 c d o t 3 6 .
Apply the exponent rule a m c d o t a n = a m + n : 3 4 + 6 .
Simplify the exponent: 3 10 .
The expression 81 c d o t 3 6 as an exponent with the base of 3 is 3 10 .
Explanation
Understanding the Problem We are given the expression 81 c d o t 3 6 and asked to write it as an exponent with the base of 3.
Expressing 81 as a Power of 3 First, we need to express 81 as a power of 3. We know that 81 = 3 c d o t 3 c d o t 3 c d o t 3 = 3 4 .
Rewriting the Expression Now we can rewrite the original expression as 3 4 c d o t 3 6 .
Simplifying the Expression Using the property of exponents that states a m c d o t a n = a m + n , we can simplify the expression: 3 4 c d o t 3 6 = 3 4 + 6 = 3 10 .
Final Answer Therefore, the expression 81 c d o t 3 6 can be written as 3 10 .
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with binary numbers (base 2) or in finance when calculating compound interest. For example, if you invest $100 at an annual interest rate of 5%, compounded annually, the amount you'll have after 10 years can be calculated using exponents: 100 c d o t ( 1 + 0.05 ) 10 . This shows how exponents help model growth and change over time.
