Answer :
Sure, let's go through each monomial one by one to determine which are perfect squares.
A monomial is a perfect square if both its numerical coefficient and the exponent of any variable are perfect squares.
1. 1:
- The number 1 is a perfect square because [tex]\(1 \times 1 = 1\)[/tex].
- There is no variable to consider.
- So, 1 is a perfect square.
2. 24:
- To be a perfect square, the number must be the square of some integer.
- The square root of 24 is about 4.9, which is not an integer.
- Therefore, 24 is not a perfect square.
3. [tex]\(66x\)[/tex]:
- First, check the numerical coefficient: 66.
- The square root of 66 is about 8.1, which is not an integer.
- Then, check the exponent of [tex]\(x\)[/tex]: [tex]\(x\)[/tex] (which can be assumed to be [tex]\(x^1\)[/tex]).
- The exponent 1 is not a perfect square.
- Therefore, [tex]\(66x\)[/tex] is not a perfect square.
4. [tex]\(49x^2\)[/tex]:
- First, check the numerical coefficient: 49.
- The square root of 49 is 7, which is an integer.
- Then, check the exponent of [tex]\(x\)[/tex]: 2.
- The square root of 2 is about 1.4, which is not an integer, but 2 itself is a perfect square in terms of exponents (since [tex]\(2 = 1 \times 2\)[/tex]).
- Therefore, [tex]\(49x^2\)[/tex] is a perfect square.
5. [tex]\(100x^3\)[/tex]:
- First, check the numerical coefficient: 100.
- The square root of 100 is 10, which is an integer.
- Then, check the exponent of [tex]\(x\)[/tex]: 3.
- The square root of 3 is about 1.7, which is not an integer, and 3 itself is not a perfect square.
- Therefore, [tex]\(100x^3\)[/tex] is not a perfect square.
6. [tex]\(81x^6\)[/tex]:
- First, check the numerical coefficient: 81.
- The square root of 81 is 9, which is an integer.
- Then, check the exponent of [tex]\(x\)[/tex]: 6.
- The square root of 6 is about 2.4, which is not an integer, but 6 itself is a perfect square in terms of exponents (since [tex]\(6 = 2 \times 3\)[/tex]).
- Therefore, [tex]\(81x^6\)[/tex] is a perfect square.
So, the monomials that are perfect squares from the given list are:
1. 1
2. [tex]\(49x^2\)[/tex]
3. [tex]\(81x^6\)[/tex]
A monomial is a perfect square if both its numerical coefficient and the exponent of any variable are perfect squares.
1. 1:
- The number 1 is a perfect square because [tex]\(1 \times 1 = 1\)[/tex].
- There is no variable to consider.
- So, 1 is a perfect square.
2. 24:
- To be a perfect square, the number must be the square of some integer.
- The square root of 24 is about 4.9, which is not an integer.
- Therefore, 24 is not a perfect square.
3. [tex]\(66x\)[/tex]:
- First, check the numerical coefficient: 66.
- The square root of 66 is about 8.1, which is not an integer.
- Then, check the exponent of [tex]\(x\)[/tex]: [tex]\(x\)[/tex] (which can be assumed to be [tex]\(x^1\)[/tex]).
- The exponent 1 is not a perfect square.
- Therefore, [tex]\(66x\)[/tex] is not a perfect square.
4. [tex]\(49x^2\)[/tex]:
- First, check the numerical coefficient: 49.
- The square root of 49 is 7, which is an integer.
- Then, check the exponent of [tex]\(x\)[/tex]: 2.
- The square root of 2 is about 1.4, which is not an integer, but 2 itself is a perfect square in terms of exponents (since [tex]\(2 = 1 \times 2\)[/tex]).
- Therefore, [tex]\(49x^2\)[/tex] is a perfect square.
5. [tex]\(100x^3\)[/tex]:
- First, check the numerical coefficient: 100.
- The square root of 100 is 10, which is an integer.
- Then, check the exponent of [tex]\(x\)[/tex]: 3.
- The square root of 3 is about 1.7, which is not an integer, and 3 itself is not a perfect square.
- Therefore, [tex]\(100x^3\)[/tex] is not a perfect square.
6. [tex]\(81x^6\)[/tex]:
- First, check the numerical coefficient: 81.
- The square root of 81 is 9, which is an integer.
- Then, check the exponent of [tex]\(x\)[/tex]: 6.
- The square root of 6 is about 2.4, which is not an integer, but 6 itself is a perfect square in terms of exponents (since [tex]\(6 = 2 \times 3\)[/tex]).
- Therefore, [tex]\(81x^6\)[/tex] is a perfect square.
So, the monomials that are perfect squares from the given list are:
1. 1
2. [tex]\(49x^2\)[/tex]
3. [tex]\(81x^6\)[/tex]
