On many calculators, there is a yx button. Depending on the calculators entry mode (algebraic or RPN) will determine how to calculate this. You might put the first number (the base to be raised) then push yx, then 4 [the exponent], then the equals sign.
If it is RPN, then enter the base number, then press Enter, then the exponent {4 in this case}, then press the yx button. Alternatively, to raise a number to the 4th power, you could just press the x² button twice. Raising to the power of 2, then raising again to power 2, is the same as raising to the power of 4.
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