Question

kuta software - infinite algebra 1
name
simplifying radical expressions
date
simplify.

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

Explanation:

Step1: Simplify \(\sqrt{125n}\)

Factor 125 into \(25\times5\), so \(\sqrt{125n}=\sqrt{25\times5n}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we get \(\sqrt{25}\times\sqrt{5n}=5\sqrt{5n}\).

Step2: Simplify \(\sqrt{216v}\)

Factor 216 into \(36\times6\), so \(\sqrt{216v}=\sqrt{36\times6v}\). Then \(\sqrt{36}\times\sqrt{6v}=6\sqrt{6v}\).

Step3: Simplify \(\sqrt{512k^{2}}\)

Factor 512 into \(256\times2\), so \(\sqrt{512k^{2}}=\sqrt{256\times2\times k^{2}}\). Using the property, we have \(\sqrt{256}\times\sqrt{k^{2}}\times\sqrt{2}=16|k|\sqrt{2}\). If \(k\geq0\), it can be written as \(16k\sqrt{2}\).

Step4: Simplify \(\sqrt{512m^{3}}\)

Factor 512 into \(256\times2\) and \(m^{3}\) into \(m^{2}\times m\). So \(\sqrt{512m^{3}}=\sqrt{256\times2\times m^{2}\times m}\). Then \(\sqrt{256}\times\sqrt{m^{2}}\times\sqrt{2m}=16m\sqrt{2m}\) (assuming \(m\geq0\)).

Step5: Simplify \(\sqrt{216k^{4}}\)

Factor 216 into \(36\times6\) and \(k^{4}=(k^{2})^{2}\). So \(\sqrt{216k^{4}}=\sqrt{36\times6\times(k^{2})^{2}}\). Then \(\sqrt{36}\times\sqrt{(k^{2})^{2}}\times\sqrt{6}=6k^{2}\sqrt{6}\).

Step6: Simplify \(\sqrt{100v^{3}}\)

Factor 100 into \(100\) and \(v^{3}=v^{2}\times v\). So \(\sqrt{100v^{3}}=\sqrt{100\times v^{2}\times v}\). Then \(\sqrt{100}\times\sqrt{v^{2}}\times\sqrt{v}=10v\sqrt{v}\) (assuming \(v\geq0\)).

Step7: Simplify \(\sqrt{80p^{3}}\)

Factor 80 into \(16\times5\) and \(p^{3}=p^{2}\times p\). So \(\sqrt{80p^{3}}=\sqrt{16\times5\times p^{2}\times p}\). Then \(\sqrt{16}\times\sqrt{p^{2}}\times\sqrt{5p}=4p\sqrt{5p}\) (assuming \(p\geq0\)).

Step8: Simplify \(\sqrt{45p^{2}}\)

Factor 45 into \(9\times5\), so \(\sqrt{45p^{2}}=\sqrt{9\times5\times p^{2}}\). Then \(\sqrt{9}\times\sqrt{p^{2}}\times\sqrt{5}=3|p|\sqrt{5}\). If \(p\geq0\), it is \(3p\sqrt{5}\).

Step9: Simplify \(\sqrt{147m^{3}n^{3}}\)

Factor 147 into \(49\times3\), \(m^{3}=m^{2}\times m\), \(n^{3}=n^{2}\times n\). So \(\sqrt{147m^{3}n^{3}}=\sqrt{49\times3\times m^{2}\times m\times n^{2}\times n}\). Then \(\sqrt{49}\times\sqrt{m^{2}}\times\sqrt{n^{2}}\times\sqrt{3mn}=7mn\sqrt{3mn}\) (assuming \(m\geq0,n\geq0\)).

Step10: Simplify \(\sqrt{200m^{4}n}\)

Factor 200 into \(100\times2\) and \(m^{4}=(m^{2})^{2}\). So \(\sqrt{200m^{4}n}=\sqrt{100\times2\times(m^{2})^{2}\times n}\). Then \(\sqrt{100}\times\sqrt{(m^{2})^{2}}\times\sqrt{2n}=10m^{2}\sqrt{2n}\).

Step11: Simplify \(\sqrt{75x^{2}y}\)

Factor 75 into \(25\times3\), so \(\sqrt{75x^{2}y}=\sqrt{25\times3\times x^{2}\times y}\). Then \(\sqrt{25}\times\sqrt{x^{2}}\times\sqrt{3y}=5|x|\sqrt{3y}\). If \(x\geq0\), it is \(5x\sqrt{3y}\).

Step12: Simplify \(\sqrt{64m^{3}n^{3}}\)

Factor 64 into \(64\), \(m^{3}=m^{2}\times m\), \(n^{3}=n^{2}\times n\). So \(\sqrt{64m^{3}n^{3}}=\sqrt{64\times m^{2}\times m\times n^{2}\times n}\). Then \(\sqrt{64}\times\sqrt{m^{2}}\times\sqrt{n^{2}}\times\sqrt{mn}=8mn\sqrt{mn}\) (assuming \(m\geq0,n\geq0\)).

Step13: Simplify \(\sqrt{16n^{3}y^{3}}\)

Factor 16 into \(16\), \(n^{3}=n^{2}\times n\), \(y^{3}=y^{2}\times y\). So \(\sqrt{16n^{3}y^{3}}=\sqrt{16\times n^{2}\times n\times y^{2}\times y}\). Then \(\sqrt{16}\times\sqrt{n^{2}}\times\sqrt{y^{2}}\times\sqrt{ny}=4ny\sqrt{ny}\) (assuming \(n\geq0,y\geq0\)).

Step14: Simplify \(\sqrt{28x^{3}y^{3}}\)

Factor 28 into \(4\times7\), \(x^{3}=x^{2}\times x\), \(y^{3}=y^{2}\times y\). So \(\sqrt{28x^{3}y^{3}}=\sqrt{4\times7\times x^{2}\times x\times y^{2}\times y}\). Then \(\sqrt{4}\times\sqrt{x^{2}}\times\sqrt{y^{2}}\times\sqrt{7xy}=2xy\sqrt{7xy}\) (assuming \(x\geq0,y\geq0\)).

Answer:

s:

1. \(5\sqrt{5n}\)
2. \(6\sqrt{6v}\)
3. \(16|k|\sqrt{2}\) (or \(16k\sqrt{2}\) if \(k\geq0\))
4. \(16m\sqrt{2m}\) (assuming \(m\geq0\))
5. \(6k^{2}\sqrt{6}\)
6. \(10v\sqrt{v}\) (assuming \(v\geq0\))
7. \(4p\sqrt{5p}\) (assuming \(p\geq0\))
8. \(3|p|\sqrt{5}\) (or \(3p\sqrt{5}\) if \(p\geq0\))
9. \(7mn\sqrt{3mn}\) (assuming \(m\geq0,n\geq0\))
10. \(10m^{2}\sqrt{2n}\)
11. \(5|x|\sqrt{3y}\) (or \(5x\sqrt{3y}\) if \(x\geq0\))
12. \(8mn\sqrt{mn}\) (assuming \(m\geq0,n\geq0\))
13. \(4ny\sqrt{ny}\) (assuming \(n\geq0,y\geq0\))
14. \(2xy\sqrt{7xy}\) (assuming \(x\geq0,y\geq0\))