QuestionSimplify the expression: $\sqrt{75}+2 \sqrt{27}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to simplify the expression in surds form. . A surd is a square root or other root symbol.
3. We can simplify surds by factoring the number under the root into its prime factors.

STEP 2

First, we need to simplify each surd separately. Let's start with $\sqrt{75}$ . We can do this by finding the prime factorization of75.

STEP 3

The prime factorization of75 is $3 \times5^2$ . So, we can write $\sqrt{75}$ as $\sqrt{3 \times5^2}$ .

STEP 4

We can simplify the surd by taking the square root of $^2$ out of the square root.
$\sqrt{75} = \sqrt{3 \times^2} =\sqrt{3}$

STEP 5

Next, we need to simplify the surd $2\sqrt{27}$ . We can do this by finding the prime factorization of27.

STEP 6

The prime factorization of27 is $3^3$ . So, we can write $2\sqrt{27}$ as $2\sqrt{3^3}$ .

STEP 7

We can simplify the surd by taking the square root of $3^2$ out of the square root.
$2\sqrt{27} =2\sqrt{3^3} =2 \times3\sqrt{3} =6\sqrt{3}$

STEP 8

Now that we have simplified each surd, we can add them together.
$\sqrt{75} +2\sqrt{27} =5\sqrt{3} +6\sqrt{3}$

STEP 9

Since the square root parts are the same, we can add the coefficients together.
$5\sqrt{3} +6\sqrt{3} =11\sqrt{3}$ So, $\sqrt{75}+2 \sqrt{27}$ simplifies to $11\sqrt{3}$ .

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QuestionSimplify the expression: $\sqrt{75}+2 \sqrt{27}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to simplify the expression in surds form. . A surd is a square root or other root symbol.
3. We can simplify surds by factoring the number under the root into its prime factors.

STEP 2

First, we need to simplify each surd separately. Let's start with $\sqrt{75}$ . We can do this by finding the prime factorization of75.

STEP 3

The prime factorization of75 is $3 \times5^2$ . So, we can write $\sqrt{75}$ as $\sqrt{3 \times5^2}$ .

STEP 4

We can simplify the surd by taking the square root of $^2$ out of the square root.
$\sqrt{75} = \sqrt{3 \times^2} =\sqrt{3}$

STEP 5

Next, we need to simplify the surd $2\sqrt{27}$ . We can do this by finding the prime factorization of27.

STEP 6

The prime factorization of27 is $3^3$ . So, we can write $2\sqrt{27}$ as $2\sqrt{3^3}$ .

STEP 7

We can simplify the surd by taking the square root of $3^2$ out of the square root.
$2\sqrt{27} =2\sqrt{3^3} =2 \times3\sqrt{3} =6\sqrt{3}$

STEP 8

Now that we have simplified each surd, we can add them together.
$\sqrt{75} +2\sqrt{27} =5\sqrt{3} +6\sqrt{3}$

STEP 9

Since the square root parts are the same, we can add the coefficients together.
$5\sqrt{3} +6\sqrt{3} =11\sqrt{3}$ So, $\sqrt{75}+2 \sqrt{27}$ simplifies to $11\sqrt{3}$ .

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QuestionSimplify the expression: 75+227\sqrt{75}+2 \sqrt{27}75​+227​.

Studdy Solution

STEP 1

Assumptions1. We are asked to simplify the expression in surds form. . A surd is a square root or other root symbol.3. We can simplify surds by factoring the number under the root into its prime factors.

STEP 2

First, we need to simplify each surd separately. Let's start with 75\sqrt{75}75​. We can do this by finding the prime factorization of75.

STEP 3

The prime factorization of75 is 3×523 \times5^23×52. So, we can write 75\sqrt{75}75​ as 3×52\sqrt{3 \times5^2}3×52​.

STEP 4

We can simplify the surd by taking the square root of 2^22 out of the square root.75=3×2=3\sqrt{75} = \sqrt{3 \times^2} =\sqrt{3}75​=3×2​=3​

STEP 5

Next, we need to simplify the surd 2272\sqrt{27}227​. We can do this by finding the prime factorization of27.

STEP 6

The prime factorization of27 is 333^333. So, we can write 2272\sqrt{27}227​ as 2332\sqrt{3^3}233​.

STEP 7

We can simplify the surd by taking the square root of 323^232 out of the square root.227=233=2×33=632\sqrt{27} =2\sqrt{3^3} =2 \times3\sqrt{3} =6\sqrt{3}227​=233​=2×33​=63​

STEP 8

Now that we have simplified each surd, we can add them together.75+227=53+63\sqrt{75} +2\sqrt{27} =5\sqrt{3} +6\sqrt{3}75​+227​=53​+63​

STEP 9

Since the square root parts are the same, we can add the coefficients together.53+63=1135\sqrt{3} +6\sqrt{3} =11\sqrt{3}53​+63​=113​So, 75+227\sqrt{75}+2 \sqrt{27}75​+227​ simplifies to 11311\sqrt{3}113​.

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QuestionSimplify the expression: $\sqrt{75}+2 \sqrt{27}$ .

Assumptions1. We are asked to simplify the expression in surds form. . A surd is a square root or other root symbol.
3. We can simplify surds by factoring the number under the root into its prime factors.

First, we need to simplify each surd separately. Let's start with $\sqrt{75}$ . We can do this by finding the prime factorization of75.

The prime factorization of75 is $3 \times5^2$ . So, we can write $\sqrt{75}$ as $\sqrt{3 \times5^2}$ .

We can simplify the surd by taking the square root of $^2$ out of the square root.
$\sqrt{75} = \sqrt{3 \times^2} =\sqrt{3}$

Next, we need to simplify the surd $2\sqrt{27}$ . We can do this by finding the prime factorization of27.

The prime factorization of27 is $3^3$ . So, we can write $2\sqrt{27}$ as $2\sqrt{3^3}$ .

We can simplify the surd by taking the square root of $3^2$ out of the square root.
$2\sqrt{27} =2\sqrt{3^3} =2 \times3\sqrt{3} =6\sqrt{3}$

Now that we have simplified each surd, we can add them together.
$\sqrt{75} +2\sqrt{27} =5\sqrt{3} +6\sqrt{3}$

Since the square root parts are the same, we can add the coefficients together.
$5\sqrt{3} +6\sqrt{3} =11\sqrt{3}$ So, $\sqrt{75}+2 \sqrt{27}$ simplifies to $11\sqrt{3}$ .