Raising an index to an index

When we have a number, and we raise it to a power, then raise the result to the power of another number, we multiply the powers. This is much more easily shown using algebra:

(a^x)^y=a^{xy}

Note: if a^x above is not surrounded with brackets, then this does not work, as it is then the power itself that is being raised to the second power.

Example: simplify (a^4)^7

Example: simplify (x^3)^2

Example: evaluate (2^3)^4

Example: evaluate (2^4)^2

flashcards

QuestionAnswer
What happens when a number is raised to a power, and the result is then raised to another power?Multiply the powers: (a^x)^y = a^{xy}
Why is (a^x)^y not equal to a^{x^y}?Without brackets, a^{x^y} means the power x is raised to y, not the base.
How do you simplify (a^4)^7?a^{4 \times 7} = a^{28}
How do you simplify (x^3)^2?x^{3 \times 2} = x^6
How do you evaluate (2^3)^4?2^{3 \times 4} = 2^{12} = 4096
How do you evaluate (2^4)^2?2^{4 \times 2} = 2^8 = 256