Solved on Dec 14, 2023

Find the expression equivalent to $9 a^{1 / 2} b^{14} c^{3 / 2}$ . Options: A) $\sqrt{3 a b^{28} c^{3}}$ , B) $\sqrt{3 a b^{7} c^{3}}$ , C) $\sqrt{81 a b^{28} c^{3}}$ , D) $\sqrt{81 a b^{7} c^{3}}$ .

STEP 1

Assumptions
1. We are given the expression $9 a^{1 / 2} b^{14} c^{3 / 2}$ .
2. We are looking for an equivalent expression among the provided options (A, B, C, D).
3. We need to manipulate the given expression to match one of the options.

STEP 2

First, we need to recognize that the given expression can be rewritten by breaking down the number 9 into a square of a number, since we are looking for an expression under a square root.
$9 = 3^2$

STEP 3

Rewrite the given expression using the square of 3.
$9 a^{1 / 2} b^{14} c^{3 / 2} = (3^2) a^{1 / 2} b^{14} c^{3 / 2}$

STEP 4

Now, we can express the entire expression as a square root, since $3^2$ is under the square root as well.
$9 a^{1 / 2} b^{14} c^{3 / 2} = \sqrt{(3^2)^2 a b^{28} c^3}$

STEP 5

Simplify the expression inside the square root by recognizing that $(3^2)^2 = 3^4$ .
$\sqrt{(3^2)^2 a b^{28} c^3} = \sqrt{3^4 a b^{28} c^3}$

STEP 6

Now, we can simplify $3^4$ under the square root, since $\sqrt{3^4} = 3^2$ .
$\sqrt{3^4 a b^{28} c^3} = \sqrt{(3^2)^2 a b^{28} c^3}$

STEP 7

Recognize that we can take $3^2$ out of the square root, since it is a perfect square.
$\sqrt{(3^2)^2 a b^{28} c^3} = 3^2 \sqrt{a b^{28} c^3}$

STEP 8

Since $3^2 = 9$ , we can rewrite the expression as:
$3^2 \sqrt{a b^{28} c^3} = 9 \sqrt{a b^{28} c^3}$

STEP 9

Now, we compare the expression we have derived with the options given:
$9 \sqrt{a b^{28} c^3}$
We can see that this matches option C, which is:
$\sqrt{81 a b^{28} c^{3}}$

STEP 10

Therefore, the equivalent expression to $9 a^{1 / 2} b^{14} c^{3 / 2}$ is:
$\sqrt{81 a b^{28} c^{3}}$
The correct answer is option C.

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Find the expression equivalent to $9 a^{1 / 2} b^{14} c^{3 / 2}$ . Options: A) $\sqrt{3 a b^{28} c^{3}}$ , B) $\sqrt{3 a b^{7} c^{3}}$ , C) $\sqrt{81 a b^{28} c^{3}}$ , D) $\sqrt{81 a b^{7} c^{3}}$ .

STEP 1

Assumptions
1. We are given the expression $9 a^{1 / 2} b^{14} c^{3 / 2}$ .
2. We are looking for an equivalent expression among the provided options (A, B, C, D).
3. We need to manipulate the given expression to match one of the options.

STEP 2

First, we need to recognize that the given expression can be rewritten by breaking down the number 9 into a square of a number, since we are looking for an expression under a square root.
$9 = 3^2$

STEP 3

Rewrite the given expression using the square of 3.
$9 a^{1 / 2} b^{14} c^{3 / 2} = (3^2) a^{1 / 2} b^{14} c^{3 / 2}$

STEP 4

Now, we can express the entire expression as a square root, since $3^2$ is under the square root as well.
$9 a^{1 / 2} b^{14} c^{3 / 2} = \sqrt{(3^2)^2 a b^{28} c^3}$

STEP 5

Simplify the expression inside the square root by recognizing that $(3^2)^2 = 3^4$ .
$\sqrt{(3^2)^2 a b^{28} c^3} = \sqrt{3^4 a b^{28} c^3}$

STEP 6

Now, we can simplify $3^4$ under the square root, since $\sqrt{3^4} = 3^2$ .
$\sqrt{3^4 a b^{28} c^3} = \sqrt{(3^2)^2 a b^{28} c^3}$

STEP 7

Recognize that we can take $3^2$ out of the square root, since it is a perfect square.
$\sqrt{(3^2)^2 a b^{28} c^3} = 3^2 \sqrt{a b^{28} c^3}$

STEP 8

Since $3^2 = 9$ , we can rewrite the expression as:
$3^2 \sqrt{a b^{28} c^3} = 9 \sqrt{a b^{28} c^3}$

STEP 9

Now, we compare the expression we have derived with the options given:
$9 \sqrt{a b^{28} c^3}$
We can see that this matches option C, which is:
$\sqrt{81 a b^{28} c^{3}}$

STEP 10

Therefore, the equivalent expression to $9 a^{1 / 2} b^{14} c^{3 / 2}$ is:
$\sqrt{81 a b^{28} c^{3}}$
The correct answer is option C.

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Find the expression equivalent to 9a1/2b14c3/29 a^{1 / 2} b^{14} c^{3 / 2}9a1/2b14c3/2. Options: A) 3ab28c3\sqrt{3 a b^{28} c^{3}}3ab28c3​, B) 3ab7c3\sqrt{3 a b^{7} c^{3}}3ab7c3​, C) 81ab28c3\sqrt{81 a b^{28} c^{3}}81ab28c3​, D) 81ab7c3\sqrt{81 a b^{7} c^{3}}81ab7c3​.

STEP 1

Assumptions1. We are given the expression 9a1/2b14c3/29 a^{1 / 2} b^{14} c^{3 / 2}9a1/2b14c3/2.2. We are looking for an equivalent expression among the provided options (A, B, C, D).3. We need to manipulate the given expression to match one of the options.

STEP 2

First, we need to recognize that the given expression can be rewritten by breaking down the number 9 into a square of a number, since we are looking for an expression under a square root.9=329 = 3^29=32

STEP 3

Rewrite the given expression using the square of 3.9a1/2b14c3/2=(32)a1/2b14c3/29 a^{1 / 2} b^{14} c^{3 / 2} = (3^2) a^{1 / 2} b^{14} c^{3 / 2}9a1/2b14c3/2=(32)a1/2b14c3/2

STEP 4

Now, we can express the entire expression as a square root, since 323^232 is under the square root as well.9a1/2b14c3/2=(32)2ab28c39 a^{1 / 2} b^{14} c^{3 / 2} = \sqrt{(3^2)^2 a b^{28} c^3}9a1/2b14c3/2=(32)2ab28c3​

STEP 5

Simplify the expression inside the square root by recognizing that (32)2=34(3^2)^2 = 3^4(32)2=34.(32)2ab28c3=34ab28c3\sqrt{(3^2)^2 a b^{28} c^3} = \sqrt{3^4 a b^{28} c^3}(32)2ab28c3​=34ab28c3​

STEP 6

Now, we can simplify 343^434 under the square root, since 34=32\sqrt{3^4} = 3^234​=32.34ab28c3=(32)2ab28c3\sqrt{3^4 a b^{28} c^3} = \sqrt{(3^2)^2 a b^{28} c^3}34ab28c3​=(32)2ab28c3​

STEP 7

Recognize that we can take 323^232 out of the square root, since it is a perfect square.(32)2ab28c3=32ab28c3\sqrt{(3^2)^2 a b^{28} c^3} = 3^2 \sqrt{a b^{28} c^3}(32)2ab28c3​=32ab28c3​

STEP 8

Since 32=93^2 = 932=9, we can rewrite the expression as:32ab28c3=9ab28c33^2 \sqrt{a b^{28} c^3} = 9 \sqrt{a b^{28} c^3}32ab28c3​=9ab28c3​

STEP 9

Now, we compare the expression we have derived with the options given:9ab28c39 \sqrt{a b^{28} c^3}9ab28c3​We can see that this matches option C, which is:81ab28c3\sqrt{81 a b^{28} c^{3}}81ab28c3​

STEP 10

Therefore, the equivalent expression to 9a1/2b14c3/29 a^{1 / 2} b^{14} c^{3 / 2}9a1/2b14c3/2 is:81ab28c3\sqrt{81 a b^{28} c^{3}}81ab28c3​The correct answer is option C.

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Find the expression equivalent to $9 a^{1 / 2} b^{14} c^{3 / 2}$ . Options: A) $\sqrt{3 a b^{28} c^{3}}$ , B) $\sqrt{3 a b^{7} c^{3}}$ , C) $\sqrt{81 a b^{28} c^{3}}$ , D) $\sqrt{81 a b^{7} c^{3}}$ .

Assumptions
1. We are given the expression $9 a^{1 / 2} b^{14} c^{3 / 2}$ .
2. We are looking for an equivalent expression among the provided options (A, B, C, D).
3. We need to manipulate the given expression to match one of the options.

First, we need to recognize that the given expression can be rewritten by breaking down the number 9 into a square of a number, since we are looking for an expression under a square root.
$9 = 3^2$

Rewrite the given expression using the square of 3.
$9 a^{1 / 2} b^{14} c^{3 / 2} = (3^2) a^{1 / 2} b^{14} c^{3 / 2}$

Now, we can express the entire expression as a square root, since $3^2$ is under the square root as well.
$9 a^{1 / 2} b^{14} c^{3 / 2} = \sqrt{(3^2)^2 a b^{28} c^3}$

Simplify the expression inside the square root by recognizing that $(3^2)^2 = 3^4$ .
$\sqrt{(3^2)^2 a b^{28} c^3} = \sqrt{3^4 a b^{28} c^3}$

Now, we can simplify $3^4$ under the square root, since $\sqrt{3^4} = 3^2$ .
$\sqrt{3^4 a b^{28} c^3} = \sqrt{(3^2)^2 a b^{28} c^3}$

Recognize that we can take $3^2$ out of the square root, since it is a perfect square.
$\sqrt{(3^2)^2 a b^{28} c^3} = 3^2 \sqrt{a b^{28} c^3}$

Since $3^2 = 9$ , we can rewrite the expression as:
$3^2 \sqrt{a b^{28} c^3} = 9 \sqrt{a b^{28} c^3}$

Now, we compare the expression we have derived with the options given:
$9 \sqrt{a b^{28} c^3}$
We can see that this matches option C, which is:
$\sqrt{81 a b^{28} c^{3}}$

Therefore, the equivalent expression to $9 a^{1 / 2} b^{14} c^{3 / 2}$ is:
$\sqrt{81 a b^{28} c^{3}}$
The correct answer is option C.