Question Add and subtract the given radicals. Choose the correct result or indicate if radicals cannot be combined. A. $3 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75}=6 \sqrt{18}-\sqrt{8}+9 \sqrt{2}-\sqrt{75}$ B. The radicals cannot be combined.

Studdy Solution

STEP 1

Assumptions
1. We are working with real numbers.
2. We can combine like radical terms.
3. The terms are simplified before combining.

STEP 2

First, we need to simplify each radical term by factoring out perfect squares from under the radicals.

STEP 3

Simplify $\sqrt{18}$ by factoring out the perfect square.
$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

STEP 4

Simplify $\sqrt{8}$ by factoring out the perfect square.
$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

STEP 5

Simplify $\sqrt{72}$ by factoring out the perfect square.
$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

STEP 6

Simplify $\sqrt{75}$ by factoring out the perfect square.
$\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$

STEP 7

Now, substitute the simplified radical forms back into the original expression.
$3 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75} = 3(3\sqrt{2}) - 2\sqrt{2} + 3(6\sqrt{2}) - 5\sqrt{3}$

STEP 8

Multiply the coefficients by the simplified radicals.
$3(3\sqrt{2}) = 9\sqrt{2}$ $3(6\sqrt{2}) = 18\sqrt{2}$

STEP 9

Now, rewrite the expression with the multiplied coefficients.
$9\sqrt{2} - 2\sqrt{2} + 18\sqrt{2} - 5\sqrt{3}$

STEP 10

Combine like radical terms. Note that we can only combine terms with the same radical part.
$9\sqrt{2} - 2\sqrt{2} + 18\sqrt{2} = (9 - 2 + 18)\sqrt{2}$

STEP 11

Perform the arithmetic operation inside the parentheses.
$(9 - 2 + 18) = 25$

STEP 12

Now, write down the combined like terms.
$25\sqrt{2} - 5\sqrt{3}$

STEP 13

Since there are no more like terms to combine, this is the simplified form of the expression.
The correct choice is A, and the expression is:
$3 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75} = 25\sqrt{2} - 5\sqrt{3}$

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Question Add and subtract the given radicals. Choose the correct result or indicate if radicals cannot be combined. A. $3 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75}=6 \sqrt{18}-\sqrt{8}+9 \sqrt{2}-\sqrt{75}$ B. The radicals cannot be combined.

Studdy Solution

STEP 1

Assumptions
1. We are working with real numbers.
2. We can combine like radical terms.
3. The terms are simplified before combining.

STEP 2

First, we need to simplify each radical term by factoring out perfect squares from under the radicals.

STEP 3

Simplify $\sqrt{18}$ by factoring out the perfect square.
$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

STEP 4

Simplify $\sqrt{8}$ by factoring out the perfect square.
$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

STEP 5

Simplify $\sqrt{72}$ by factoring out the perfect square.
$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

STEP 6

Simplify $\sqrt{75}$ by factoring out the perfect square.
$\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$

STEP 7

Now, substitute the simplified radical forms back into the original expression.
$3 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75} = 3(3\sqrt{2}) - 2\sqrt{2} + 3(6\sqrt{2}) - 5\sqrt{3}$

STEP 8

Multiply the coefficients by the simplified radicals.
$3(3\sqrt{2}) = 9\sqrt{2}$ $3(6\sqrt{2}) = 18\sqrt{2}$

STEP 9

Now, rewrite the expression with the multiplied coefficients.
$9\sqrt{2} - 2\sqrt{2} + 18\sqrt{2} - 5\sqrt{3}$

STEP 10

Combine like radical terms. Note that we can only combine terms with the same radical part.
$9\sqrt{2} - 2\sqrt{2} + 18\sqrt{2} = (9 - 2 + 18)\sqrt{2}$

STEP 11

Perform the arithmetic operation inside the parentheses.
$(9 - 2 + 18) = 25$

STEP 12

Now, write down the combined like terms.
$25\sqrt{2} - 5\sqrt{3}$

STEP 13

Since there are no more like terms to combine, this is the simplified form of the expression.
The correct choice is A, and the expression is:
$3 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75} = 25\sqrt{2} - 5\sqrt{3}$

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Studdy Solution

STEP 1

Assumptions1. We are working with real numbers.2. We can combine like radical terms.3. The terms are simplified before combining.

STEP 2

First, we need to simplify each radical term by factoring out perfect squares from under the radicals.

STEP 3

Simplify 18\sqrt{18}18​ by factoring out the perfect square.18=9×2=9×2=32\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}18​=9×2​=9​×2​=32​

STEP 4

Simplify 8\sqrt{8}8​ by factoring out the perfect square.8=4×2=4×2=22\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}8​=4×2​=4​×2​=22​

STEP 5

Simplify 72\sqrt{72}72​ by factoring out the perfect square.72=36×2=36×2=62\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}72​=36×2​=36​×2​=62​

STEP 6

Simplify 75\sqrt{75}75​ by factoring out the perfect square.75=25×3=25×3=53\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}75​=25×3​=25​×3​=53​

STEP 7

Now, substitute the simplified radical forms back into the original expression.318−8+372−75=3(32)−22+3(62)−533 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75} = 3(3\sqrt{2}) - 2\sqrt{2} + 3(6\sqrt{2}) - 5\sqrt{3}318​−8​+372​−75​=3(32​)−22​+3(62​)−53​

STEP 8

Multiply the coefficients by the simplified radicals.3(32)=923(3\sqrt{2}) = 9\sqrt{2}3(32​)=92​ 3(62)=1823(6\sqrt{2}) = 18\sqrt{2}3(62​)=182​

STEP 9

Now, rewrite the expression with the multiplied coefficients.92−22+182−539\sqrt{2} - 2\sqrt{2} + 18\sqrt{2} - 5\sqrt{3}92​−22​+182​−53​

STEP 10

Combine like radical terms. Note that we can only combine terms with the same radical part.92−22+182=(9−2+18)29\sqrt{2} - 2\sqrt{2} + 18\sqrt{2} = (9 - 2 + 18)\sqrt{2}92​−22​+182​=(9−2+18)2​

STEP 11

Perform the arithmetic operation inside the parentheses.(9−2+18)=25(9 - 2 + 18) = 25(9−2+18)=25

STEP 12

Now, write down the combined like terms.252−5325\sqrt{2} - 5\sqrt{3}252​−53​

STEP 13

Since there are no more like terms to combine, this is the simplified form of the expression.The correct choice is A, and the expression is:318−8+372−75=252−533 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75} = 25\sqrt{2} - 5\sqrt{3}318​−8​+372​−75​=252​−53​

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Assumptions
1. We are working with real numbers.
2. We can combine like radical terms.
3. The terms are simplified before combining.

Simplify $\sqrt{18}$ by factoring out the perfect square.
$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

Simplify $\sqrt{8}$ by factoring out the perfect square.
$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

Simplify $\sqrt{72}$ by factoring out the perfect square.
$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

Simplify $\sqrt{75}$ by factoring out the perfect square.
$\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$

Now, substitute the simplified radical forms back into the original expression.
$3 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75} = 3(3\sqrt{2}) - 2\sqrt{2} + 3(6\sqrt{2}) - 5\sqrt{3}$

Multiply the coefficients by the simplified radicals.
$3(3\sqrt{2}) = 9\sqrt{2}$ $3(6\sqrt{2}) = 18\sqrt{2}$

Now, rewrite the expression with the multiplied coefficients.
$9\sqrt{2} - 2\sqrt{2} + 18\sqrt{2} - 5\sqrt{3}$

Combine like radical terms. Note that we can only combine terms with the same radical part.
$9\sqrt{2} - 2\sqrt{2} + 18\sqrt{2} = (9 - 2 + 18)\sqrt{2}$

Perform the arithmetic operation inside the parentheses.
$(9 - 2 + 18) = 25$

Now, write down the combined like terms.
$25\sqrt{2} - 5\sqrt{3}$

Since there are no more like terms to combine, this is the simplified form of the expression.
The correct choice is A, and the expression is:
$3 \sqrt{18}-\sqrt{8}+3 \sqrt{72}-\sqrt{75} = 25\sqrt{2} - 5\sqrt{3}$