The Value of 2(256)^-1/8

To find the value of 2(256)^-1/8, we need to understand the concept of exponents and how to simplify expressions with fractional exponents.

Understanding Exponents
An exponent indicates how many times a number, called the base, should be multiplied by itself. For example, in the expression 2^3, the base is 2 and the exponent is 3. This means that we need to multiply 2 by itself three times: 2 × 2 × 2 = 8.

Simplifying Fractional Exponents
Fractional exponents are a way of representing roots. For example, the expression 4^(1/2) represents the square root of 4, which is 2. Similarly, the expression 8^(1/3) represents the cube root of 8, which is 2.

To simplify an expression with a fractional exponent, we can use the property of exponents that states a^(m/n) is equal to the nth root of a raised to the power of m. In other words, a^(m/n) = (n√a)^m.

Simplifying 2(256)^-1/8
Let's break down the expression step by step:

Step 1: Simplify the exponent
The exponent in the expression is -1/8. To simplify this, we need to take the reciprocal of the base and change the sign of the exponent. So, (256)^-1/8 is equal to 1/(256^(1/8)).

Step 2: Calculate the eighth root of 256
Now, we need to find the eighth root of 256. The eighth root of a number is the number that, when raised to the power of 8, gives the original number. In this case, 256^(1/8) is equal to 2.

Step 3: Substitute the value back into the expression
Substituting the value of 256^(1/8) back into the expression, we have 1/2.

Step 4: Multiply by 2
Finally, we multiply the result by 2. 2 * (1/2) equals 1.

The value of 2(256)^-1/8 is 1.

In summary, we simplified the expression by using the rules of exponents and fractional exponents. By taking the reciprocal of the base and changing the sign of the exponent, we converted the expression into a simpler form. Then, by finding the eighth root of 256 and substituting it back into the expression, we arrived at the final value of 1.