Which expression is equivalent to the quantity seven raised to the negative second power times three raised to the fifth power end quantity all raised to the negative second power?
a) seven raised to the fourth power divided by three raised to the tenth power
b) negative seven raised to the fourth power divided by three raised to the tenth power
c) three raised to the third power divided by seven raised to the fourth power
d) negative three raised to the third power divided by seven raised to the fourth power
To simplify the expression, we start by raising $7^{-2} \cdot 3^5$ to the $-2$nd power. We have $(7^{-2} \cdot 3^5)^{-2} = 7^{-2\cdot -2} \cdot 3^{5 \cdot -2} = 7^{4} \cdot 3^{-10} = \frac{7^4}{3^{10}}$. Thus, the answer is $\boxed{\textbf{(a)}\ seven\,\mathrm{raised\,to\,the\,fourth\,power\,divided\,by\,three\,\mathrm{raised\,to\,the}\,tenth\,\mathrm{power}}}$.
To simplify the given expression, we can start by applying the exponent rules.
The expression is: (7^-2 * 3^5)^-2
To solve this, we can multiply the exponents by each other.
First, let's multiply the exponents inside the parentheses: (7^-2 * 3^5) gives us 7^-2 * 3^5 = (1/7^2) * 3^5 = 1/49 * 243 = 243/49
Now we have: (243/49)^-2
To deal with a negative exponent, we can flip the fraction and make the exponent positive:
(243/49)^-2 = (49/243)^2 = 49^2 / 243^2 = 2401 / 59049
So the expression is equivalent to the fraction 2401/59049.
None of the given answer choices match this fraction, so none of the options provided are equivalent to the given expression.
To find the equivalent expression, let's break it down step by step.
First, we have the quantity "seven raised to the negative second power." This means we take the reciprocal of seven raised to the second power. In other words, it is equal to 1 divided by seven squared, or 1/49.
Next, we have "three raised to the fifth power." This is simply three multiplied by itself five times, which equals 243.
Now, let's raise the expression 1/49 times 243 to the negative second power. To do this, we multiply both 1/49 and 243, and then we raise the result to the negative second power.
(1/49) * 243 = 243/49
To raise this to the negative second power, we take the reciprocal and then square it. The reciprocal of 243/49 is 49/243, and squaring it gives us (49/243)^2.
So, the expression is equivalent to (49/243)^2.
Now, let's compare the answer choices:
a) seven raised to the fourth power divided by three raised to the tenth power.
b) negative seven raised to the fourth power divided by three raised to the tenth power.
c) three raised to the third power divided by seven raised to the fourth power.
d) negative three raised to the third power divided by seven raised to the fourth power.
None of the answer choices match our equivalent expression (49/243)^2. Therefore, the correct answer is none of the above.