Let's solve this problem step-by-step.
### Part a) Is the number a perfect square?
We are given the number:
[tex]\[
2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 11
\][/tex]
First, let's express this number in terms of its prime factors:
[tex]\[
2^4 \times 7^2 \times 11
\][/tex]
A number is a perfect square if all the exponents in its prime factorization are even. Let’s examine the exponents of the prime factors in this expression:
- The exponent of 2 is 4, which is even.
- The exponent of 7 is 2, which is even.
- The exponent of 11 is 1, which is odd.
Since the exponent of 11 is odd, the number is not a perfect square.
### Part b) What is the least whole number you could multiply it by to get a perfect square?
To make the number a perfect square, we need to make all exponents in its prime factorization even. Currently, the exponent of 11 is 1 (odd). To make it even, we need to multiply by another 11. This will give us:
[tex]\[
2^4 \times 7^2 \times 11 \times 11 = 2^4 \times 7^2 \times 11^2
\][/tex]
Now, all the exponents (4 for 2, 2 for 7, and 2 for 11) are even, making the number a perfect square.
Therefore, we need to multiply the original number by 11.
### Part c) What is the square root of the perfect square from part b)?
The number we obtained after multiplying by 11 is:
[tex]\[
2^4 \times 7^2 \times 11^2
\][/tex]
To find the square root of this perfect square, we take the square root of each factor:
[tex]\[
\sqrt{2^4 \times 7^2 \times 11^2} = 2^2 \times 7 \times 11
\][/tex]
Calculating this:
[tex]\[
2^2 = 4
\][/tex]
[tex]\[
4 \times 7 = 28
\][/tex]
[tex]\[
28 \times 11 = 308
\][/tex]
Thus, the square root of the perfect square is 308.
### Summary
So, the results for each part of the question are:
- a) The number [tex]\( 2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 11 \)[/tex] is not a perfect square.
- b) To make it a perfect square, you need to multiply it by 11.
- c) The square root of the resulting perfect square is 308.
