Math

QuestionSimplify 405\sqrt{405} into the format aba \sqrt{b}. What is 405=\sqrt{405}=?

Studdy Solution

STEP 1

Assumptions1. We need to simplify 405\sqrt{405} to the form aba \sqrt{b}. . Here, aa and bb are integers and bb is as small as possible.

STEP 2

We start by finding the prime factorization of405.

STEP 3

We can start by dividing405 by the smallest prime number, which is2. Since405 is not divisible by2, we try the next prime number, which is3.
405÷3=135405 \div3 =135

STEP 4

We continue this process with the quotient135. It is also divisible by3.
135÷3=45135 \div3 =45

STEP 5

We continue this process with the quotient45. It is also divisible by3.
45÷3=1545 \div3 =15

STEP 6

We continue this process with the quotient15. It is also divisible by3.
15÷3=515 \div3 =5

STEP 7

Finally, we divide5 by the smallest prime number that it is divisible by, which is5.
5÷5=15 \div5 =1

STEP 8

So, the prime factorization of405 is 34×53^4 \times5.

STEP 9

Now, we use the property of square roots that says ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}.
405=34×5=34×5\sqrt{405} = \sqrt{3^4 \times5} = \sqrt{3^4} \times \sqrt{5}

STEP 10

We simplify 34\sqrt{3^4} to 323^2.
405=32×5=95\sqrt{405} =3^2 \times \sqrt{5} =9\sqrt{5}So, a=9a =9 and b=5b =5.

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