Solved on Feb 01, 2024

Simplify the product of $\sqrt{18}$ and $\sqrt{63}$ .

STEP 1

Assumptions
1. We need to simplify the product of two square roots.
2. The properties of square roots will be used, including the fact that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ .

STEP 2

We will use the property of square roots that allows us to multiply the radicands (the numbers inside the square roots) together.
$\sqrt{18} \cdot \sqrt{63} = \sqrt{18 \cdot 63}$

STEP 3

Calculate the product of the radicands.
$18 \cdot 63$

STEP 4

Perform the multiplication.
$18 \cdot 63 = 1134$

STEP 5

Now we have the square root of the product.
$\sqrt{18} \cdot \sqrt{63} = \sqrt{1134}$

STEP 6

Factor the radicand into its prime factors to simplify the square root.
$1134 = 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 7$

STEP 7

Group the prime factors into pairs of equal factors, if possible, to simplify the square root. Each pair of equal factors can come out of the square root as a single factor.
$\sqrt{1134} = \sqrt{2 \cdot 3^4 \cdot 7}$

STEP 8

Since $3^4 = (3^2)^2 = 9^2$ , we can take $3^2 = 9$ out of the square root.
$\sqrt{1134} = \sqrt{2 \cdot 9^2 \cdot 7}$

STEP 9

Take the square of 9 out of the square root.
$\sqrt{1134} = 9\sqrt{2 \cdot 7}$

STEP 10

Multiply the remaining factors inside the square root.
$\sqrt{1134} = 9\sqrt{14}$

STEP 11

Write the final simplified product.
$\sqrt{18} \cdot \sqrt{63} = 9\sqrt{14}$
The product in its simplest form is $9\sqrt{14}$ .

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Simplify the product of $\sqrt{18}$ and $\sqrt{63}$ .

STEP 1

Assumptions
1. We need to simplify the product of two square roots.
2. The properties of square roots will be used, including the fact that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ .

STEP 2

We will use the property of square roots that allows us to multiply the radicands (the numbers inside the square roots) together.
$\sqrt{18} \cdot \sqrt{63} = \sqrt{18 \cdot 63}$

STEP 3

Calculate the product of the radicands.
$18 \cdot 63$

STEP 4

Perform the multiplication.
$18 \cdot 63 = 1134$

STEP 5

Now we have the square root of the product.
$\sqrt{18} \cdot \sqrt{63} = \sqrt{1134}$

STEP 6

Factor the radicand into its prime factors to simplify the square root.
$1134 = 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 7$

STEP 7

Group the prime factors into pairs of equal factors, if possible, to simplify the square root. Each pair of equal factors can come out of the square root as a single factor.
$\sqrt{1134} = \sqrt{2 \cdot 3^4 \cdot 7}$

STEP 8

Since $3^4 = (3^2)^2 = 9^2$ , we can take $3^2 = 9$ out of the square root.
$\sqrt{1134} = \sqrt{2 \cdot 9^2 \cdot 7}$

STEP 9

Take the square of 9 out of the square root.
$\sqrt{1134} = 9\sqrt{2 \cdot 7}$

STEP 10

Multiply the remaining factors inside the square root.
$\sqrt{1134} = 9\sqrt{14}$

STEP 11

Write the final simplified product.
$\sqrt{18} \cdot \sqrt{63} = 9\sqrt{14}$
The product in its simplest form is $9\sqrt{14}$ .

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Simplify the product of 18\sqrt{18}18​ and 63\sqrt{63}63​.

STEP 1

Assumptions1. We need to simplify the product of two square roots.2. The properties of square roots will be used, including the fact that a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}a​⋅b​=ab​.

STEP 2

We will use the property of square roots that allows us to multiply the radicands (the numbers inside the square roots) together.18⋅63=18⋅63\sqrt{18} \cdot \sqrt{63} = \sqrt{18 \cdot 63}18​⋅63​=18⋅63​

STEP 3

Calculate the product of the radicands.18⋅6318 \cdot 6318⋅63

STEP 4

Perform the multiplication.18⋅63=113418 \cdot 63 = 113418⋅63=1134

STEP 5

Now we have the square root of the product.18⋅63=1134\sqrt{18} \cdot \sqrt{63} = \sqrt{1134}18​⋅63​=1134​

STEP 6

Factor the radicand into its prime factors to simplify the square root.1134=2⋅3⋅3⋅3⋅3⋅71134 = 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 71134=2⋅3⋅3⋅3⋅3⋅7

STEP 7

Group the prime factors into pairs of equal factors, if possible, to simplify the square root. Each pair of equal factors can come out of the square root as a single factor.1134=2⋅34⋅7\sqrt{1134} = \sqrt{2 \cdot 3^4 \cdot 7}1134​=2⋅34⋅7​

STEP 8

Since 34=(32)2=923^4 = (3^2)^2 = 9^234=(32)2=92, we can take 32=93^2 = 932=9 out of the square root.1134=2⋅92⋅7\sqrt{1134} = \sqrt{2 \cdot 9^2 \cdot 7}1134​=2⋅92⋅7​

STEP 9

Take the square of 9 out of the square root.1134=92⋅7\sqrt{1134} = 9\sqrt{2 \cdot 7}1134​=92⋅7​

STEP 10

Multiply the remaining factors inside the square root.1134=914\sqrt{1134} = 9\sqrt{14}1134​=914​

STEP 11

Write the final simplified product.18⋅63=914\sqrt{18} \cdot \sqrt{63} = 9\sqrt{14}18​⋅63​=914​The product in its simplest form is 9149\sqrt{14}914​.

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Simplify the product of $\sqrt{18}$ and $\sqrt{63}$ .

Assumptions
1. We need to simplify the product of two square roots.
2. The properties of square roots will be used, including the fact that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ .

We will use the property of square roots that allows us to multiply the radicands (the numbers inside the square roots) together.
$\sqrt{18} \cdot \sqrt{63} = \sqrt{18 \cdot 63}$

Calculate the product of the radicands.
$18 \cdot 63$

Perform the multiplication.
$18 \cdot 63 = 1134$

Now we have the square root of the product.
$\sqrt{18} \cdot \sqrt{63} = \sqrt{1134}$

Factor the radicand into its prime factors to simplify the square root.
$1134 = 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 7$

Group the prime factors into pairs of equal factors, if possible, to simplify the square root. Each pair of equal factors can come out of the square root as a single factor.
$\sqrt{1134} = \sqrt{2 \cdot 3^4 \cdot 7}$

Since $3^4 = (3^2)^2 = 9^2$ , we can take $3^2 = 9$ out of the square root.
$\sqrt{1134} = \sqrt{2 \cdot 9^2 \cdot 7}$

Take the square of 9 out of the square root.
$\sqrt{1134} = 9\sqrt{2 \cdot 7}$

Multiply the remaining factors inside the square root.
$\sqrt{1134} = 9\sqrt{14}$

Write the final simplified product.
$\sqrt{18} \cdot \sqrt{63} = 9\sqrt{14}$
The product in its simplest form is $9\sqrt{14}$ .