What is 1/2 to the 4th power as a fraction?
The Correct Answer and Explanation is :
To calculate ( \left(\frac{1}{2}\right)^4 ), we can follow the rules of exponents. When we raise a fraction to a power, we raise both the numerator and the denominator to that power. Thus, we can express this as follows:
[
\left(\frac{1}{2}\right)^4 = \frac{1^4}{2^4}
]
Next, we calculate ( 1^4 ) and ( 2^4 ):
- ( 1^4 = 1 ) (because any number raised to any power remains the same when the number is one).
- ( 2^4 = 2 \times 2 \times 2 \times 2 ).
Calculating ( 2^4 ):
- ( 2 \times 2 = 4 )
- ( 4 \times 2 = 8 )
- ( 8 \times 2 = 16 )
So, ( 2^4 = 16 ).
Now, substituting these values back into the fraction, we have:
[
\left(\frac{1}{2}\right)^4 = \frac{1}{16}
]
Thus, ( \left(\frac{1}{2}\right)^4 ) simplifies to ( \frac{1}{16} ).
Explanation
The process of raising a fraction to a power involves understanding the properties of exponents. The key rule to remember is that ( (a/b)^n = a^n / b^n ). This means that the numerator and denominator are both raised to the same exponent.
In this case, the base fraction ( \frac{1}{2} ) has a numerator of 1 and a denominator of 2. By applying the exponent of 4, we calculated ( 1^4 ) and ( 2^4 ) separately.
Raising a number to the power of 4 involves multiplying that number by itself four times. For example, ( 2^4 ) is calculated through sequential multiplication, which helps in understanding the concept of powers and the growth of numbers as they are multiplied.
The result, ( \frac{1}{16} ), indicates that when the fraction ( \frac{1}{2} ) is multiplied by itself four times, it results in a much smaller value, demonstrating how exponential decay works in fractions. Hence, the final answer is:
[
\left(\frac{1}{2}\right)^4 = \frac{1}{16}
]