(8^8)^3
To solve (8^8)^3, we need to apply the exponentiation rules.
First, we solve the exponent inside the parentheses:
8^8 = 8 x 8 x 8 x 8 x 8 x 8 x 8 x 8 = 16,777,216.
Now, we raise this result to the power of 3:
(8^8)^3 = 16,777,216^3 = 4,973,686,472,081,026,816.
Therefore, (8^8)^3 equals 4,973,686,472,081,026,816.
please simplify that
To simplify (8^8)^3, we can apply the exponentiation rule which states that raising an exponent to another exponent is equivalent to multiplying the exponents.
Therefore, (8^8)^3 simplifies to 8^(8*3).
Now, we can multiply the exponents: 8*3 = 24.
So, (8^8)^3 simplifies to 8^24.
4^−2⋅4⋅4^−4=
To simplify the expression 4^(-2) ⋅ 4 ⋅ 4^(-4), we'll use the rules of exponents.
Recall that a negative exponent can be rewritten as the reciprocal of the number raised to the positive exponent. Hence, 4^(-2) = 1/4^2 and 4^(-4) = 1/4^4.
Now, we can substitute these values back into the expression:
(1/4^2) ⋅ 4 ⋅ (1/4^4)
Next, we simplify the exponents:
(1/16) ⋅ 4 ⋅ (1/256)
Multiply the numbers:
1/16 ⋅ 4/1 ⋅ 1/256 = (4/16) ⋅ (1/256)
Reduce fractions if possible:
1/4 ⋅ 1/256
Multiply the numbers:
1/1024
Therefore, 4^(-2) ⋅ 4 ⋅ 4^(-4) simplifies to 1/1024.
fully simplify that please
To fully simplify 4^(-2) ⋅ 4 ⋅ 4^(-4), we can apply the rules of exponents.
Recall that a negative exponent can be rewritten as the reciprocal of the number raised to the positive exponent. Thus, 4^(-2) = 1/4^2 and 4^(-4) = 1/4^4.
Now, we substitute these values back into the expression:
(1/4^2) ⋅ 4 ⋅ (1/4^4)
Next, we simplify the exponents:
(1/16) ⋅ 4 ⋅ (1/256)
We can multiply the numbers:
1/16 × 4 × 1/256 = 1/(16 × 256)
Further simplifying:
1/(4,096)
Therefore, 4^(-2) ⋅ 4 ⋅ 4^(-4) simplified to 1/4,096.
fully simplfy
\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}
To fully simplify the expression (1/5) ⋅ (1/5) ⋅ (1/5) ⋅ (1/5), we can multiply the numerators and the denominators together:
Numerator: 1 ⋅ 1 ⋅ 1 ⋅ 1 = 1
Denominator: 5 ⋅ 5 ⋅ 5 ⋅ 5 = 625
Therefore, (1/5) ⋅ (1/5) ⋅ (1/5) ⋅ (1/5) simplifies to 1/625.
\left(\frac{y^{5}}{y^{2}}\right)^{2}\cdot\left(\frac{y^{7}}{y^{6}}\right)^{5}
To simplify the expression:
\left(\frac{y^{5}}{y^{2}}\right)^{2}\cdot\left(\frac{y^{7}}{y^{6}}\right)^{5}
We can apply the exponentiation rule which states that raising a power to another power is equivalent to multiplying the exponents.
Inside the first set of parentheses:
\left(\frac{y^{5}}{y^{2}}\right)^{2} becomes \frac{y^{5 \cdot 2}}{y^{2 \cdot 2}} which simplifies to \frac{y^{10}}{y^{4}}.
Inside the second set of parentheses:
\left(\frac{y^{7}}{y^{6}}\right)^{5} becomes \frac{y^{7 \cdot 5}}{y^{6 \cdot 5}} which simplifies to \frac{y^{35}}{y^{30}}.
Now, we can multiply the two expressions:
\frac{y^{10}}{y^{4}} \cdot \frac{y^{35}}{y^{30}}.
To multiply the expressions, we can subtract the exponents when the base is the same:
y^{10 - 4} \cdot y^{35 - 30} = y^{6} \cdot y^{5} = y^{6 + 5} = y^{11}.
Therefore, \left(\frac{y^{5}}{y^{2}}\right)^{2}\cdot\left(\frac{y^{7}}{y^{6}}\right)^{5} simplifies to y^{11}.
\left(x^{3}\right)^{9}\cdot x^{3}
To fully simplify the expression:
\left(x^{3}\right)^{9}\cdot x^{3}
We can apply the exponentiation rule which states that raising a power to another power is equivalent to multiplying the exponents.
Inside the first set of parentheses:
\left(x^{3}\right)^{9} becomes x^{3 \cdot 9} which simplifies to x^{27}.
Now, we can multiply x^{27} by x^{3}, applying the rule that when multiplying with the same base, we add the exponents:
x^{27} \cdot x^{3} = x^{27 + 3} = x^{30}.
Therefore, \left(x^{3}\right)^{9}\cdot x^{3} simplifies to x^{30}.
\left(x^{-2}\cdot x^{8}\right)^{9}
To simplify the expression:
\left(x^{-2} \cdot x^{8}\right)^{9}
We can multiply the exponents inside the parentheses:
x^{-2} \cdot x^{8} = x^{(-2) + 8} = x^{6}
Now, we can raise the result to the power of 9:
\left(x^{6}\right)^{9} = x^{6 \cdot 9} = x^{54}
Therefore, \left(x^{-2} \cdot x^{8}\right)^{9} simplifies to x^{54}.
\frac{\left(x^{-6}\cdot x^{3}\right)^{3}}{\left(x^{4}\cdot x^{3}\right)^{3}}
To simplify the expression:
\frac{\left(x^{-6} \cdot x^{3}\right)^{3}}{\left(x^{4} \cdot x^{3}\right)^{3}}
We can simplify each set of parentheses individually before dividing the results.
Inside the first set of parentheses:
x^{-6} \cdot x^{3} = x^{(-6) + 3} = x^{-3}
Inside the second set of parentheses:
x^{4} \cdot x^{3} = x^{4 + 3} = x^{7}
Now, we can substitute the simplified expressions back into the original expression:
\frac{\left(x^{-3}\right)^{3}}{\left(x^{7}\right)^{3}}
Applying the exponentiation rule to each term:
\frac{x^{(-3) \cdot 3}}{x^{7 \cdot 3}}
Simplifying the exponents:
\frac{x^{-9}}{x^{21}}
To divide with the same base, we subtract the exponents:
x^{-9 - 21} = x^{-30}
Therefore, \frac{\left(x^{-6} \cdot x^{3}\right)^{3}}{\left(x^{4} \cdot x^{3}\right)^{3}} simplifies to x^{-30}.
Students were given the task to fully simplify:
\left(2^{2}\cdot2^{-6}\right)^{-3}\cdot\left(2\right)^{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\times2=4
and for the exponents 2+-6=-4 and -4\times-3=12
What is John's error?
What is Carol's error?
What is the correct answer. Show your work.
John's error lies in assuming that any number raised to the power of zero equals 1. While this is true for most numbers, including 2 in this case, it is not true for 0 raised to the power of zero, which is undefined.
Carol's error lies in her calculation of the exponents. While she correctly calculates 2^2 as 4 and 2^(-6) as 1/64, her mistake is that she adds the exponents in the parentheses, resulting in -4. However, the correct operation in this case is multiplication, not addition.
To find the correct answer, let's simplify the expression step by step:
First, inside the first set of parentheses, we can simplify:
2^2 ⋅ 2^(-6) = 4 ⋅ 1/64 = 1/16.
Now, we can rewrite the expression as:
(1/16)^(-3) ⋅ 2^0.
Applying the exponentiation rule, we have:
(1/16)^(-3) = (16/1)^3 = 1^3 = 1.
Finally, we have:
1 ⋅ 2^0 = 1 ⋅ 1 = 1.
Therefore, the correct answer is indeed 1.