QuestionSimplify: $3 \sqrt{2} + \sqrt{32}$

Studdy Solution

STEP 1

Assumptions1. We are working with real numbers. We are using the standard rules of algebra and arithmetic3. The square root symbol represents the principal square root, which is always positive or zero

STEP 2

We need to simplify the term $\sqrt{32}$ . We can do this by finding the prime factorization of32 and then simplifying the square root.

STEP 3

Find the prime factorization of32.
$32 =2^5$

STEP 4

Substitute the prime factorization into the square root.
\sqrt{32} = \sqrt{2^}

STEP 5

Use the rule of square roots, which states that $\sqrt{a^b} = a^{b/2}$ , to simplify the square root.
$\sqrt{2^5} =2^{5/2}$

STEP 6

implify the exponent by dividing5 by2.
$2^{5/2} =2^{2.5} =2^2 \cdot2^{0.5} =4 \cdot \sqrt{2}$

STEP 7

Substitute the simplified form of $\sqrt{32}$ back into the original expression.
$3 \sqrt{2}+\sqrt{32} =3 \sqrt{2} +4 \sqrt{2}$

STEP 8

Combine like terms.
$3 \sqrt{2} +4 \sqrt{2} =7 \sqrt{2}$ So, the simplified form of the expression $3 \sqrt{2}+\sqrt{32}$ is $7 \sqrt{2}$ .

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QuestionSimplify: $3 \sqrt{2} + \sqrt{32}$

Studdy Solution

STEP 1

Assumptions1. We are working with real numbers. We are using the standard rules of algebra and arithmetic3. The square root symbol represents the principal square root, which is always positive or zero

STEP 2

We need to simplify the term $\sqrt{32}$ . We can do this by finding the prime factorization of32 and then simplifying the square root.

STEP 3

Find the prime factorization of32.
$32 =2^5$

STEP 4

Substitute the prime factorization into the square root.
\sqrt{32} = \sqrt{2^}

STEP 5

Use the rule of square roots, which states that $\sqrt{a^b} = a^{b/2}$ , to simplify the square root.
$\sqrt{2^5} =2^{5/2}$

STEP 6

implify the exponent by dividing5 by2.
$2^{5/2} =2^{2.5} =2^2 \cdot2^{0.5} =4 \cdot \sqrt{2}$

STEP 7

Substitute the simplified form of $\sqrt{32}$ back into the original expression.
$3 \sqrt{2}+\sqrt{32} =3 \sqrt{2} +4 \sqrt{2}$

STEP 8

Combine like terms.
$3 \sqrt{2} +4 \sqrt{2} =7 \sqrt{2}$ So, the simplified form of the expression $3 \sqrt{2}+\sqrt{32}$ is $7 \sqrt{2}$ .

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QuestionSimplify: 32+323 \sqrt{2} + \sqrt{32}32​+32​

Studdy Solution

STEP 1

Assumptions1. We are working with real numbers. We are using the standard rules of algebra and arithmetic3. The square root symbol represents the principal square root, which is always positive or zero

STEP 2

We need to simplify the term 32\sqrt{32}32​. We can do this by finding the prime factorization of32 and then simplifying the square root.

STEP 3

Find the prime factorization of32.32=2532 =2^532=25

STEP 4

Substitute the prime factorization into the square root.\sqrt{32} = \sqrt{2^}

STEP 5

Use the rule of square roots, which states that ab=ab/2\sqrt{a^b} = a^{b/2}ab​=ab/2, to simplify the square root.25=25/2\sqrt{2^5} =2^{5/2}25​=25/2

STEP 6

implify the exponent by dividing5 by2.25/2=22.5=22⋅20.5=4⋅22^{5/2} =2^{2.5} =2^2 \cdot2^{0.5} =4 \cdot \sqrt{2}25/2=22.5=22⋅20.5=4⋅2​

STEP 7

Substitute the simplified form of 32\sqrt{32}32​ back into the original expression.32+32=32+423 \sqrt{2}+\sqrt{32} =3 \sqrt{2} +4 \sqrt{2}32​+32​=32​+42​

STEP 8

Combine like terms.32+42=723 \sqrt{2} +4 \sqrt{2} =7 \sqrt{2}32​+42​=72​So, the simplified form of the expression 32+323 \sqrt{2}+\sqrt{32}32​+32​ is 727 \sqrt{2}72​.

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QuestionSimplify: $3 \sqrt{2} + \sqrt{32}$

We need to simplify the term $\sqrt{32}$ . We can do this by finding the prime factorization of32 and then simplifying the square root.

Find the prime factorization of32.
$32 =2^5$

Substitute the prime factorization into the square root.
\sqrt{32} = \sqrt{2^}

Use the rule of square roots, which states that $\sqrt{a^b} = a^{b/2}$ , to simplify the square root.
$\sqrt{2^5} =2^{5/2}$

implify the exponent by dividing5 by2.
$2^{5/2} =2^{2.5} =2^2 \cdot2^{0.5} =4 \cdot \sqrt{2}$

Substitute the simplified form of $\sqrt{32}$ back into the original expression.
$3 \sqrt{2}+\sqrt{32} =3 \sqrt{2} +4 \sqrt{2}$

Combine like terms.
$3 \sqrt{2} +4 \sqrt{2} =7 \sqrt{2}$ So, the simplified form of the expression $3 \sqrt{2}+\sqrt{32}$ is $7 \sqrt{2}$ .