Final answer:
The student's question relates to exponentiation, where a number is multiplied by itself a specified number of times. For 3^3, it is 27; 4^7 is a larger number reached by multiplying 4 by itself seven times; and 5^6 is 5 multiplied by itself six times, resulting in 15625.
Step-by-step explanation:
Raising a Number to a Power
When we raise a number to a power, we are performing an operation known as exponentiation. For example, 3^3 or 3 x 3 x 3 is the number 3 multiplied by itself three times, which equals 27. Similarly, raising 4 to the power of 7, written as 4^7, involves multiplying 4 by itself seven times. The process is like taking the number (in this case, 4) and using it as a factor repeatedly in a multiplication process, which saves space compared to writing out every individual multiplication.
For 5^6, we multiply the number 5 by itself six times, which can be thought of as (5×5×5×5×5×5). This exponentiation illustrates how quickly numbers can grow when using powers, as 5 raised to the 6th power is equal to 15625.
The ability to work with powers is a fundamental skill in mathematics that can help you understand the nature of exponential growth and how to manipulate expressions that include exponents. The concept of exponentiation applies to both integer exponents, like those shown in the student's question, as well as non-integer exponents, where a number is raised to a fractional or decimal power.