Students were given the task to fully simplify
(2^2*2^-6)^-3*(2)^0
John believes the answer was 1 as any number to the zero power (except 0^0 ) equals 1.
Carol believes that the answer is 4^12 since
2*2=4 and for the exponents 2+-6=-4 and -4*-3=12
What is John's error?
What is Carol's error?
What is the correct answer. Show your work.
Math
John's error: John incorrectly assumed that any number raised to the power of 0 is always equal to 1. However, in this specific expression, there are other factors that need to be taken into consideration.
Carol's error: Carol mistakenly simplified 2^2 to 4 and then further simplified 2^-6 to -4. Additionally, she multiplied the exponents instead of adding them.
To find the correct answer, we need to follow the order of operations (PEMDAS).
(2^2 * 2^-6)^-3 * 2^0
First, we simplify the expression inside the parentheses:
2^2 * 2^-6 = 2^(2 + -6) = 2^-4
Next, we simplify the exponent -3 with the base 2^-4:
(2^-4)^-3
Applying the power of a power rule, we multiply the two exponents:
2^((-4) * (-3)) = 2^12
Finally, we simplify 2^0:
2^12 * 2^0 = 2^12
So, the correct answer is 2^12.
John's error is in the belief that any number raised to the power of zero equals 1. This is not the case for any number other than 0. The correct rule is that any non-zero number raised to the power of zero equals 1. In this case, 2^0 does indeed equal 1, but it does not affect the rest of the expression.
Carol's error is in the simplification of the exponents. 2^2 * 2^-6 is not equal to 4^-4. Instead, when multiplying numbers with the same base, you add the exponents. Therefore, 2^2 * 2^-6 simplifies to 2^(2+(-6)) = 2^-4.
To fully simplify the expression, we can calculate each component step by step:
(2^2 * 2^-6)^-3 * 2^0
First, simplify the exponent inside the parentheses:
2^2 * 2^-6 = 2^(2+(-6)) = 2^-4
Now, integrate the simplified expression into the larger expression:
(2^-4)^-3 * 2^0
Next, apply the power of a power property, which states that when raising a power to another power, you multiply the exponents:
2^((-4)(-3)) * 2^0
Simplify the exponent:
2^12 * 2^0
Lastly, any number raised to the power of zero equals 1:
2^12 * 1
And the final answer is:
2^12 = 4096
John's error is that he is applying the rule for any number (except 0) raised to the power of 0. While it is true that any non-zero number raised to the power of 0 equals 1, the expression being simplified here involves an exponent that is a product of other exponents, which requires a different approach.
Carol's error is in failing to simplify the exponent before applying the power rule. The expression (2^2*2^-6)^-3 can be simplified before applying the rule of any number to the power of 0.
To solve this problem correctly, we need to simplify the expression step by step:
First, let's simplify the exponent (2^2*2^-6)^-3:
Using the product of powers rule, we can combine the exponents inside the parentheses:
2^2 * 2^-6 = 2^(2 + (-6)) = 2^(-4)
Now, we have (2^(-4))^-3:
Using the power of a power rule, we multiply the exponents:
(2^(-4))^-3 = 2^((-4)*(-3)) = 2^12
Finally, we have 2^12 * 2^0:
Using the rule of any number to the power of 0, we know that 2^0 equals 1, so we have:
2^12 * 1 = 2^12
Therefore, the correct answer is 2^12, which means Carol was correct in her calculation.