Math / Algebra

Questionsimplify. 1) $\sqrt{125 n}$ 2) $\sqrt{216 v}$ 3) $\sqrt{512 k^{2}}$ 4) $\sqrt{512 m^{3}}$ 5) $\sqrt{216 k^{4}}$ 6) $\sqrt{100 v^{3}}$ 7) $\sqrt{80 p^{3}}$ 8) $\sqrt{45 p^{2}}$ 9) $\sqrt{147 m^{3} n^{3}}$ 10) $\sqrt{200 m^{4} n}$ 11) $\sqrt{75 x^{2} y}$ 12) $\sqrt{64 m^{3} n^{3}}$ 13) $\sqrt{16 u^{4} v^{3}}$ 14) $\sqrt{28 x^{3} y^{3}}$

Studdy Solution

STEP 1

1. We are asked to simplify each square root expression.
2. Simplification involves factoring out perfect squares from under the square root.

STEP 2

1. Factor the number and variable expressions under each square root.
2. Identify and extract perfect squares from each expression.
3. Simplify the square root by taking the square root of the perfect squares.

STEP 3

For $\sqrt{125n}$ :
1. Factor 125 into $5^2 \times 5$ .
2. The expression becomes $\sqrt{5^2 \times 5 \times n}$ .

STEP 4

Extract the perfect square:
1. $\sqrt{5^2 \times 5 \times n} = 5\sqrt{5n}$ .

STEP 5

For $\sqrt{216v}$ :
1. Factor 216 into $6^3$ or $2^3 \times 3^3$ .
2. The expression becomes $\sqrt{6^2 \times 6 \times v}$ .

STEP 6

Extract the perfect square:
1. $\sqrt{6^2 \times 6 \times v} = 6\sqrt{6v}$ .

STEP 7

For $\sqrt{512k^2}$ :
1. Factor 512 into $2^9$ .
2. The expression becomes $\sqrt{2^8 \times 2 \times k^2}$ .

STEP 8

Extract the perfect square:
1. $\sqrt{2^8 \times 2 \times k^2} = 16k\sqrt{2}$ .

STEP 9

For $\sqrt{512m^3}$ :
1. Factor 512 into $2^9$ .
2. The expression becomes $\sqrt{2^8 \times 2 \times m^2 \times m}$ .

STEP 10

Extract the perfect square:
1. $\sqrt{2^8 \times 2 \times m^2 \times m} = 16m\sqrt{2m}$ .

STEP 11

For $\sqrt{216k^4}$ :
1. Factor 216 into $6^3$ .
2. The expression becomes $\sqrt{6^2 \times 6 \times k^4}$ .

STEP 12

Extract the perfect square:
1. $\sqrt{6^2 \times 6 \times k^4} = 6k^2\sqrt{6}$ .

STEP 13

For $\sqrt{100v^3}$ :
1. Factor 100 into $10^2$ .
2. The expression becomes $\sqrt{10^2 \times v^2 \times v}$ .

STEP 14

Extract the perfect square:
1. $\sqrt{10^2 \times v^2 \times v} = 10v\sqrt{v}$ .

STEP 15

For $\sqrt{80p^3}$ :
1. Factor 80 into $4 \times 20$ or $4 \times 2^2 \times 5$ .
2. The expression becomes $\sqrt{4 \times 2^2 \times 5 \times p^2 \times p}$ .

STEP 16

Extract the perfect square:
1. $\sqrt{4 \times 2^2 \times 5 \times p^2 \times p} = 4p\sqrt{5p}$ .

STEP 17

For $\sqrt{45p^2}$ :
1. Factor 45 into $9 \times 5$ .
2. The expression becomes $\sqrt{9 \times 5 \times p^2}$ .

STEP 18

Extract the perfect square:
1. $\sqrt{9 \times 5 \times p^2} = 3p\sqrt{5}$ .

STEP 19

For $\sqrt{147m^3n^3}$ :
1. Factor 147 into $49 \times 3$ .
2. The expression becomes $\sqrt{49 \times 3 \times m^2 \times m \times n^2 \times n}$ .

STEP 20

Extract the perfect square:
1. $\sqrt{49 \times 3 \times m^2 \times m \times n^2 \times n} = 7mn\sqrt{3mn}$ .

STEP 21

For $\sqrt{200m^4n}$ :
1. Factor 200 into $100 \times 2$ .
2. The expression becomes $\sqrt{100 \times 2 \times m^4 \times n}$ .

STEP 22

Extract the perfect square:
1. $\sqrt{100 \times 2 \times m^4 \times n} = 10m^2\sqrt{2n}$ .

STEP 23

For $\sqrt{75x^2y}$ :
1. Factor 75 into $25 \times 3$ .
2. The expression becomes $\sqrt{25 \times 3 \times x^2 \times y}$ .

STEP 24

Extract the perfect square:
1. $\sqrt{25 \times 3 \times x^2 \times y} = 5x\sqrt{3y}$ .

STEP 25

For $\sqrt{64m^3n^3}$ :
1. Factor 64 into $8^2$ .
2. The expression becomes $\sqrt{8^2 \times m^2 \times m \times n^2 \times n}$ .

STEP 26

Extract the perfect square:
1. $\sqrt{8^2 \times m^2 \times m \times n^2 \times n} = 8mn\sqrt{mn}$ .

STEP 27

For $\sqrt{16u^4v^3}$ :
1. Factor 16 into $4^2$ .
2. The expression becomes $\sqrt{4^2 \times u^4 \times v^2 \times v}$ .

STEP 28

Extract the perfect square:
1. $\sqrt{4^2 \times u^4 \times v^2 \times v} = 4u^2v\sqrt{v}$ .

STEP 29

For $\sqrt{28x^3y^3}$ :
1. Factor 28 into $4 \times 7$ .
2. The expression becomes $\sqrt{4 \times 7 \times x^2 \times x \times y^2 \times y}$ .

STEP 30

Extract the perfect square:
1. $\sqrt{4 \times 7 \times x^2 \times x \times y^2 \times y} = 2xy\sqrt{7xy}$ .

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Studdy Solution

STEP 1

1. We are asked to simplify each square root expression.2. Simplification involves factoring out perfect squares from under the square root.

STEP 2

1. Factor the number and variable expressions under each square root.2. Identify and extract perfect squares from each expression.3. Simplify the square root by taking the square root of the perfect squares.

STEP 3

For 125n\sqrt{125n}125n​:1. Factor 125 into 52×55^2 \times 552×5.2. The expression becomes 52×5×n\sqrt{5^2 \times 5 \times n}52×5×n​.

STEP 4

Extract the perfect square:1. 52×5×n=55n\sqrt{5^2 \times 5 \times n} = 5\sqrt{5n}52×5×n​=55n​.

STEP 5

For 216v\sqrt{216v}216v​:1. Factor 216 into 636^363 or 23×332^3 \times 3^323×33.2. The expression becomes 62×6×v\sqrt{6^2 \times 6 \times v}62×6×v​.

STEP 6

Extract the perfect square:1. 62×6×v=66v\sqrt{6^2 \times 6 \times v} = 6\sqrt{6v}62×6×v​=66v​.

STEP 7

For 512k2\sqrt{512k^2}512k2​:1. Factor 512 into 292^929.2. The expression becomes 28×2×k2\sqrt{2^8 \times 2 \times k^2}28×2×k2​.

STEP 8

Extract the perfect square:1. 28×2×k2=16k2\sqrt{2^8 \times 2 \times k^2} = 16k\sqrt{2}28×2×k2​=16k2​.

STEP 9

For 512m3\sqrt{512m^3}512m3​:1. Factor 512 into 292^929.2. The expression becomes 28×2×m2×m\sqrt{2^8 \times 2 \times m^2 \times m}28×2×m2×m​.

STEP 10

Extract the perfect square:1. 28×2×m2×m=16m2m\sqrt{2^8 \times 2 \times m^2 \times m} = 16m\sqrt{2m}28×2×m2×m​=16m2m​.

STEP 11

For 216k4\sqrt{216k^4}216k4​:1. Factor 216 into 636^363.2. The expression becomes 62×6×k4\sqrt{6^2 \times 6 \times k^4}62×6×k4​.

STEP 12

Extract the perfect square:1. 62×6×k4=6k26\sqrt{6^2 \times 6 \times k^4} = 6k^2\sqrt{6}62×6×k4​=6k26​.

STEP 13

For 100v3\sqrt{100v^3}100v3​:1. Factor 100 into 10210^2102.2. The expression becomes 102×v2×v\sqrt{10^2 \times v^2 \times v}102×v2×v​.

STEP 14

Extract the perfect square:1. 102×v2×v=10vv\sqrt{10^2 \times v^2 \times v} = 10v\sqrt{v}102×v2×v​=10vv​.

STEP 15

For 80p3\sqrt{80p^3}80p3​:1. Factor 80 into 4×204 \times 204×20 or 4×22×54 \times 2^2 \times 54×22×5.2. The expression becomes 4×22×5×p2×p\sqrt{4 \times 2^2 \times 5 \times p^2 \times p}4×22×5×p2×p​.

STEP 16

Extract the perfect square:1. 4×22×5×p2×p=4p5p\sqrt{4 \times 2^2 \times 5 \times p^2 \times p} = 4p\sqrt{5p}4×22×5×p2×p​=4p5p​.

STEP 17

For 45p2\sqrt{45p^2}45p2​:1. Factor 45 into 9×59 \times 59×5.2. The expression becomes 9×5×p2\sqrt{9 \times 5 \times p^2}9×5×p2​.

STEP 18

Extract the perfect square:1. 9×5×p2=3p5\sqrt{9 \times 5 \times p^2} = 3p\sqrt{5}9×5×p2​=3p5​.

STEP 19

For 147m3n3\sqrt{147m^3n^3}147m3n3​:1. Factor 147 into 49×349 \times 349×3.2. The expression becomes 49×3×m2×m×n2×n\sqrt{49 \times 3 \times m^2 \times m \times n^2 \times n}49×3×m2×m×n2×n​.

STEP 20

Extract the perfect square:1. 49×3×m2×m×n2×n=7mn3mn\sqrt{49 \times 3 \times m^2 \times m \times n^2 \times n} = 7mn\sqrt{3mn}49×3×m2×m×n2×n​=7mn3mn​.

STEP 21

For 200m4n\sqrt{200m^4n}200m4n​:1. Factor 200 into 100×2100 \times 2100×2.2. The expression becomes 100×2×m4×n\sqrt{100 \times 2 \times m^4 \times n}100×2×m4×n​.

STEP 22

Extract the perfect square:1. 100×2×m4×n=10m22n\sqrt{100 \times 2 \times m^4 \times n} = 10m^2\sqrt{2n}100×2×m4×n​=10m22n​.

STEP 23

For 75x2y\sqrt{75x^2y}75x2y​:1. Factor 75 into 25×325 \times 325×3.2. The expression becomes 25×3×x2×y\sqrt{25 \times 3 \times x^2 \times y}25×3×x2×y​.

STEP 24

Extract the perfect square:1. 25×3×x2×y=5x3y\sqrt{25 \times 3 \times x^2 \times y} = 5x\sqrt{3y}25×3×x2×y​=5x3y​.

STEP 25

For 64m3n3\sqrt{64m^3n^3}64m3n3​:1. Factor 64 into 828^282.2. The expression becomes 82×m2×m×n2×n\sqrt{8^2 \times m^2 \times m \times n^2 \times n}82×m2×m×n2×n​.

STEP 26

Extract the perfect square:1. 82×m2×m×n2×n=8mnmn\sqrt{8^2 \times m^2 \times m \times n^2 \times n} = 8mn\sqrt{mn}82×m2×m×n2×n​=8mnmn​.

STEP 27

For 16u4v3\sqrt{16u^4v^3}16u4v3​:1. Factor 16 into 424^242.2. The expression becomes 42×u4×v2×v\sqrt{4^2 \times u^4 \times v^2 \times v}42×u4×v2×v​.

STEP 28

Extract the perfect square:1. 42×u4×v2×v=4u2vv\sqrt{4^2 \times u^4 \times v^2 \times v} = 4u^2v\sqrt{v}42×u4×v2×v​=4u2vv​.

STEP 29

For 28x3y3\sqrt{28x^3y^3}28x3y3​:1. Factor 28 into 4×74 \times 74×7.2. The expression becomes 4×7×x2×x×y2×y\sqrt{4 \times 7 \times x^2 \times x \times y^2 \times y}4×7×x2×x×y2×y​.

STEP 30

Extract the perfect square:1. 4×7×x2×x×y2×y=2xy7xy\sqrt{4 \times 7 \times x^2 \times x \times y^2 \times y} = 2xy\sqrt{7xy}4×7×x2×x×y2×y​=2xy7xy​.

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1. We are asked to simplify each square root expression.
2. Simplification involves factoring out perfect squares from under the square root.

1. Factor the number and variable expressions under each square root.
2. Identify and extract perfect squares from each expression.
3. Simplify the square root by taking the square root of the perfect squares.

For $\sqrt{125n}$ :
1. Factor 125 into $5^2 \times 5$ .
2. The expression becomes $\sqrt{5^2 \times 5 \times n}$ .

Extract the perfect square:
1. $\sqrt{5^2 \times 5 \times n} = 5\sqrt{5n}$ .

For $\sqrt{216v}$ :
1. Factor 216 into $6^3$ or $2^3 \times 3^3$ .
2. The expression becomes $\sqrt{6^2 \times 6 \times v}$ .

Extract the perfect square:
1. $\sqrt{6^2 \times 6 \times v} = 6\sqrt{6v}$ .

For $\sqrt{512k^2}$ :
1. Factor 512 into $2^9$ .
2. The expression becomes $\sqrt{2^8 \times 2 \times k^2}$ .

Extract the perfect square:
1. $\sqrt{2^8 \times 2 \times k^2} = 16k\sqrt{2}$ .

For $\sqrt{512m^3}$ :
1. Factor 512 into $2^9$ .
2. The expression becomes $\sqrt{2^8 \times 2 \times m^2 \times m}$ .

Extract the perfect square:
1. $\sqrt{2^8 \times 2 \times m^2 \times m} = 16m\sqrt{2m}$ .

For $\sqrt{216k^4}$ :
1. Factor 216 into $6^3$ .
2. The expression becomes $\sqrt{6^2 \times 6 \times k^4}$ .

Extract the perfect square:
1. $\sqrt{6^2 \times 6 \times k^4} = 6k^2\sqrt{6}$ .

For $\sqrt{100v^3}$ :
1. Factor 100 into $10^2$ .
2. The expression becomes $\sqrt{10^2 \times v^2 \times v}$ .

Extract the perfect square:
1. $\sqrt{10^2 \times v^2 \times v} = 10v\sqrt{v}$ .

For $\sqrt{80p^3}$ :
1. Factor 80 into $4 \times 20$ or $4 \times 2^2 \times 5$ .
2. The expression becomes $\sqrt{4 \times 2^2 \times 5 \times p^2 \times p}$ .

Extract the perfect square:
1. $\sqrt{4 \times 2^2 \times 5 \times p^2 \times p} = 4p\sqrt{5p}$ .

For $\sqrt{45p^2}$ :
1. Factor 45 into $9 \times 5$ .
2. The expression becomes $\sqrt{9 \times 5 \times p^2}$ .

Extract the perfect square:
1. $\sqrt{9 \times 5 \times p^2} = 3p\sqrt{5}$ .

For $\sqrt{147m^3n^3}$ :
1. Factor 147 into $49 \times 3$ .
2. The expression becomes $\sqrt{49 \times 3 \times m^2 \times m \times n^2 \times n}$ .

Extract the perfect square:
1. $\sqrt{49 \times 3 \times m^2 \times m \times n^2 \times n} = 7mn\sqrt{3mn}$ .

For $\sqrt{200m^4n}$ :
1. Factor 200 into $100 \times 2$ .
2. The expression becomes $\sqrt{100 \times 2 \times m^4 \times n}$ .

Extract the perfect square:
1. $\sqrt{100 \times 2 \times m^4 \times n} = 10m^2\sqrt{2n}$ .

For $\sqrt{75x^2y}$ :
1. Factor 75 into $25 \times 3$ .
2. The expression becomes $\sqrt{25 \times 3 \times x^2 \times y}$ .

Extract the perfect square:
1. $\sqrt{25 \times 3 \times x^2 \times y} = 5x\sqrt{3y}$ .

For $\sqrt{64m^3n^3}$ :
1. Factor 64 into $8^2$ .
2. The expression becomes $\sqrt{8^2 \times m^2 \times m \times n^2 \times n}$ .

Extract the perfect square:
1. $\sqrt{8^2 \times m^2 \times m \times n^2 \times n} = 8mn\sqrt{mn}$ .

For $\sqrt{16u^4v^3}$ :
1. Factor 16 into $4^2$ .
2. The expression becomes $\sqrt{4^2 \times u^4 \times v^2 \times v}$ .

Extract the perfect square:
1. $\sqrt{4^2 \times u^4 \times v^2 \times v} = 4u^2v\sqrt{v}$ .

For $\sqrt{28x^3y^3}$ :
1. Factor 28 into $4 \times 7$ .
2. The expression becomes $\sqrt{4 \times 7 \times x^2 \times x \times y^2 \times y}$ .

Extract the perfect square:
1. $\sqrt{4 \times 7 \times x^2 \times x \times y^2 \times y} = 2xy\sqrt{7xy}$ .