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

• =a^{4\times7}
• =a^{28}

Example: simplify (x^3)^2

• =x^{3\times2}
• =x^6

Example: evaluate (2^3)^4

• =2^{3\times4}
• =2^{12}
• =4096

Example: evaluate (2^4)^2

• =2^{4\times2}
• =2^8
• =256