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Multiply Square Roots

Multiply square roots

Multiply Square Roots

We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says

\[\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}\]

We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.

\[\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}\]

Remember, we assume all variables are greater than or equal to zero.

We will rewrite the Product Property of Square Roots so we see both ways together.

So we can multiply \(\sqrt{3}\cdot \sqrt{5}\) in this way:

\[\begin{array}{l}\sqrt{3}\cdot \sqrt{5} \\ \sqrt{3\cdot 5} \\ \sqrt{15}\end{array}\]

Sometimes the product gives us a perfect square:

\[\begin{array}{l}\sqrt{2}\cdot \sqrt{8} \\ \sqrt{2\cdot 8} \\ \sqrt{16} \\ 4\end{array}\]

Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.

Example

Try it.

Simplify: ⓐ \(\sqrt{2}\cdot \sqrt{6}\) ⓑ \((4\sqrt{3})(2\sqrt{12})\).

Solution

ⓐ

	\(\sqrt{2}\cdot \sqrt{6}\)
Multiply using the Product Property.	\(\sqrt{12}\)
Simplify the radical.	\(\sqrt{4}\cdot \sqrt{3}\)
Simplify.	\(2\sqrt{3}\)

ⓑ

	\((4\sqrt{3})(2\sqrt{12})\)
Multiply using the Product Property.	\(8\sqrt{36}\)
Simplify the radical.	\(8\cdot 6\)
Simplify.	\(48\)

Notice that in (b) we multiplied the coefficients and multiplied the radicals. Also, we did not simplify \(\sqrt{12}\). We waited to get the product and then simplified.

Example

Try it.

Simplify: \((6\sqrt{2})(3\sqrt{10})\).

Solution

	\((6\sqrt{2})(3\sqrt{10})\)
Multiply using the Product Property.	\(18\sqrt{20}\)
Simplify the radical.	\(18\sqrt{4}\cdot \sqrt{5}\)
Simplify.	\(18\cdot 2\cdot \sqrt{5}\)
	\(36\sqrt{5}\)

Example

Try it.

Simplify: ⓐ \((\sqrt{8{x}^{3}})(\sqrt{3x})\) ⓑ \((\sqrt{20{y}^{2}})(\sqrt{5{y}^{3}})\).

Solution

ⓐ

	\((\sqrt{8{x}^{3}})(\sqrt{3x})\)
Multiply using the Product Property.	\(\sqrt{24{x}^{4}}\)
Simplify the radical.	\(\sqrt{4{x}^{4}}\cdot \sqrt{6}\)
Simplify.	\(2{x}^{2}\sqrt{6}\)

ⓑ

	\((\sqrt{20{y}^{2}})(\sqrt{5{y}^{3}})\)
Multiply using the Product Property.	\(\sqrt{100{y}^{5}}\)
Simplify the radical.	\(10{y}^{2}\sqrt{y}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Use Polynomial Multiplication to Multiply Square Roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots.

We will first distribute and then simplify the square roots when possible.

Example

Try it.

Simplify: ⓐ \(3(5-\sqrt{2})\) ⓑ \(\sqrt{2}(4-\sqrt{10})\).

Solution

ⓐ

	\(3(5-\sqrt{2})\)
Distribute.	\(15-3\sqrt{2}\)

ⓑ

	\(\sqrt{2}(4-\sqrt{10})\)
Distribute.	\(4\sqrt{2}-\sqrt{20}\)
	\(4\sqrt{2}-2\sqrt{5}\)

Example

Try it.

Simplify: ⓐ \(\sqrt{5}(7+2\sqrt{5})\) ⓑ \(\sqrt{6}(\sqrt{2}+\sqrt{18})\).

Solution

ⓐ

	\(\sqrt{5}(7+2\sqrt{5})\)
Multiply.	\(7\sqrt{5}+2\cdot 5\)
Simplify.	\(7\sqrt{5}+10\)
	\(10+7\sqrt{5}\)

ⓑ

	\(\sqrt{6}(\sqrt{2}+\sqrt{18})\)
Multiply.	\(\sqrt{12}+\sqrt{108}\)
Simplify.	\(\sqrt{4}\cdot \sqrt{3}+\sqrt{36}\cdot \sqrt{3}\)
	\(2\sqrt{3}+6\sqrt{3}\)
Combine like radicals.	\(8\sqrt{3}\)

When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work, usually by the FOIL method.

Example

Try it.

Simplify: \((2+\sqrt{3})(4-\sqrt{3})\).

Solution

	\((2+\sqrt{3})(4-\sqrt{3})\)
Multiply.	\(8-2\sqrt{3}+4\sqrt{3}-3\)
Combine like terms.	\(5+2\sqrt{3}\)

Example

Try it.

Simplify: \((3-2\sqrt{7})(4-2\sqrt{7})\).

Solution

	\((3-2\sqrt{7})(4-2\sqrt{7})\)
Multiply.	\(12-6\sqrt{7}-8\sqrt{7}+4\cdot 7\)
Simplify.	\(12-6\sqrt{7}-8\sqrt{7}+28\)
Combine like terms.	\(40-14\sqrt{7}\)

Example

Try it.

Simplify: \((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\).

Solution

	\((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
Multiply.	\(3\cdot 2+12\sqrt{10}-\sqrt{10}-4\cdot 5\)
Simplify.	\(6+12\sqrt{10}-\sqrt{10}-20\)
Combine like terms.	\(-14+11\sqrt{10}\)

Example

Try it.

Simplify: \((4-2\sqrt{x})(1+3\sqrt{x})\).

Solution

	\((4-2\sqrt{x})(1+3\sqrt{x})\)
Multiply.	\(4+12\sqrt{x}-2\sqrt{x}-6x\)
Combine like terms.	\(4+10\sqrt{x}-6x\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

Product Property of Square Roots If a, b are nonnegative real numbers, then
\[\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}\ \text{and}\ \sqrt{a}\cdot \sqrt{b}=\sqrt{ab}\]
Special formulas for multiplying binomials and conjugates:
\[\begin{array}{llll}{(a+b)}^{2}={a}^{2}+2ab+{b}^{2} & & & (a-b)(a+b)={a}^{2}-{b}^{2} \\ {(a-b)}^{2}={a}^{2}-2ab+{b}^{2} & & & \end{array}\]
The FOIL method can be used to multiply binomials containing radicals.

Multiply Square Roots

Multiply Square Roots

In the following exercises, simplify.

Try it.

ⓐ \(\sqrt{2}\cdot \sqrt{8}\) ⓑ \((3\sqrt{3})(2\sqrt{18})\)

Solution

ⓐ \(4\) ⓑ \(18\sqrt{6}\)

Try it.

ⓐ \(\sqrt{6}\cdot \sqrt{6}\) ⓑ \((3\sqrt{2})(2\sqrt{32})\)

Try it.

ⓐ \(\sqrt{7}\cdot \sqrt{14}\) ⓑ \((4\sqrt{8})(5\sqrt{8})\)

Solution

ⓐ \(7\sqrt{2}\) ⓑ 160

Try it.

ⓐ \(\sqrt{6}\cdot \sqrt{12}\) ⓑ \((2\sqrt{5})(2\sqrt{10})\)

Try it.

\((5\sqrt{2})(3\sqrt{6})\)

Solution

\(30\sqrt{3}\)

Try it.

\((2\sqrt{3})(4\sqrt{6})\)

Try it.

\((-2\sqrt{3})(3\sqrt{18})\)

Solution

\(-18\sqrt{6}\)

Try it.

\((-4\sqrt{5})(5\sqrt{10})\)

Try it.

\((5\sqrt{6})(\text{-}\sqrt{12})\)

Solution

\(-30\sqrt{2}\)

Try it.

\((6\sqrt{2})(\text{-}\sqrt{10})\)

Try it.

\((-2\sqrt{7})(-2\sqrt{14})\)

Solution

\(28\sqrt{2}\)

Try it.

\((-2\sqrt{11})(-4\sqrt{22})\)

Try it.

ⓐ \((\sqrt{15y})(\sqrt{5{y}^{3}})\) ⓑ \((\sqrt{2{n}^{2}})(\sqrt{18{n}^{3}})\)

Solution

ⓐ \(5{y}^{2}\sqrt{3}\) ⓑ \(6{n}^{2}\sqrt{n}\)

Try it.

ⓐ \((\sqrt{14{x}^{3}})(\sqrt{7{x}^{3}})\) ⓑ \((\sqrt{3{q}^{2}})(\sqrt{48{q}^{3}})\)

Try it.

ⓐ \((\sqrt{16{y}^{2}})(\sqrt{8{y}^{4}})\) ⓑ \((\sqrt{11{s}^{6}})(\sqrt{11s})\)

Solution

ⓐ \(8{y}^{3}\sqrt{2}\) ⓑ \(11{s}^{3}\sqrt{s}\)

Try it.

ⓐ \((\sqrt{8{x}^{3}})(\sqrt{3x})\) ⓑ \((\sqrt{7r})(\sqrt{7{r}^{8}})\)

Try it.

\((2\sqrt{5{b}^{3}})(4\sqrt{15b})\)

Solution

\(40{b}^{2}\sqrt{3}\)

Try it.

\((3\sqrt{8{c}^{5}})(2\sqrt{6{c}^{3}})\)

Try it.

\((5\sqrt{2{d}^{7}})(3\sqrt{50{d}^{3}})\)

Solution

\(150{d}^{5}\)

Try it.

\(\left(4\sqrt{6{t}^{2}}\right)\left(3\sqrt{3{t}^{2}}\right)\)

Try it.

\(\left(3\sqrt{4{y}^{4}}\right)\left(3\sqrt{9{y}^{5}}\right)\)

Solution

\(54{y}^{4}\sqrt{y}\)

Try it.

\((-2\sqrt{7{z}^{3}})(3\sqrt{14{z}^{8}})\)

Try it.

\((4\sqrt{2{k}^{5}})(-3\sqrt{32{k}^{6}})\)

Solution

\(-96{k}^{5}\sqrt{k}\)

Try it.

ⓐ \({(\sqrt{7})}^{2}\) ⓑ \({(\text{-}\sqrt{15})}^{2}\)

Try it.

ⓐ \({(\sqrt{11})}^{2}\) ⓑ \({(\text{-}\sqrt{21})}^{2}\)

Solution

ⓐ 11 ⓑ 21

Try it.

ⓐ \({(\sqrt{19})}^{2}\) ⓑ \({(\text{-}\sqrt{5})}^{2}\)

Try it.

ⓐ \({(\sqrt{23})}^{2}\)
ⓑ \({(\text{-}\sqrt{3})}^{2}\)

Solution

ⓐ 23 ⓑ 3

Try it.

ⓐ \((4\sqrt{11})(-3\sqrt{11})\) ⓑ \({(5\sqrt{3})}^{2}\)

Try it.

ⓐ \((2\sqrt{13})(-9\sqrt{13})\) ⓑ \({(6\sqrt{5})}^{2}\)

Solution

ⓐ −234 ⓑ 180

Try it.

ⓐ \((-3\sqrt{12})(-2\sqrt{6})\) ⓑ \({(-4\sqrt{10})}^{2}\)

Try it.

ⓐ \((-7\sqrt{5})(-3\sqrt{10})\) ⓑ \({(-2\sqrt{14})}^{2}\)

Solution

ⓐ \(105\sqrt{2}\) ⓑ 56

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

Try it.

ⓐ \(3(4-\sqrt{3})\) ⓑ \(\sqrt{2}(4-\sqrt{6})\)

Try it.

ⓐ \(4(6-\sqrt{11})\) ⓑ \(\sqrt{2}(5-\sqrt{12})\)

Solution

ⓐ \(24-4\sqrt{11}\) ⓑ \(5\sqrt{2}-2\sqrt{6}\)

Try it.

ⓐ \(5(3-\sqrt{7})\) ⓑ \(\sqrt{3}(4-\sqrt{15})\)

Try it.

ⓐ \(7(-2-\sqrt{11})\) ⓑ \(\sqrt{7}(6-\sqrt{14})\)

Solution

ⓐ \(-14-7\sqrt{11}\) ⓑ \(6\sqrt{7}-7\sqrt{2}\)

Try it.

ⓐ \(\sqrt{7}(5+2\sqrt{7})\) ⓑ \(\sqrt{5}(\sqrt{10}+\sqrt{18})\)

Try it.

ⓐ \(\sqrt{11}(8+4\sqrt{11})\) ⓑ \(\sqrt{3}(\sqrt{12}+\sqrt{27})\)

Solution

ⓐ \(44+8\sqrt{11}\) ⓑ \(15\)

Try it.

ⓐ \(\sqrt{11}(-3+4\sqrt{11})\) ⓑ \(\sqrt{3}(\sqrt{15}-\sqrt{18})\)

Try it.

ⓐ \(\sqrt{2}(-5+9\sqrt{2})\) ⓑ \(\sqrt{7}(\sqrt{3}-\sqrt{21})\)

Solution

ⓐ \(18-5\sqrt{2}\) ⓑ \(\sqrt{21}-7\sqrt{3}\)

Try it.

\((8+\sqrt{3})(2-\sqrt{3})\)

Try it.

\((7+\sqrt{3})(9-\sqrt{3})\)

Solution

\(60+2\sqrt{3}\)

Try it.

\((8-\sqrt{2})(3+\sqrt{2})\)

Try it.

\((9-\sqrt{2})(6+\sqrt{2})\)

Solution

\(52+3\sqrt{2}\)

Try it.

\((3-\sqrt{7})(5-\sqrt{7})\)

Try it.

\((5-\sqrt{7})(4-\sqrt{7})\)

Solution

\(27-9\sqrt{7}\)

Try it.

\((1+3\sqrt{10})(5-2\sqrt{10})\)

Try it.

\((7-2\sqrt{5})(4+9\sqrt{5})\)

Solution

\(-62+55\sqrt{5}\)

Try it.

\((\sqrt{3}+\sqrt{10})(\sqrt{3}+2\sqrt{10})\)

Try it.

\((\sqrt{11}+\sqrt{5})(\sqrt{11}+6\sqrt{5})\)

Solution

\(41+7\sqrt{55}\)

Try it.

\((2\sqrt{7}-5\sqrt{11})(4\sqrt{7}+9\sqrt{11})\)

Try it.

\((4\sqrt{6}+7\sqrt{13})(8\sqrt{6}-3\sqrt{13})\)

Solution

\(-81+44\sqrt{78}\)

Try it.

\((5-\sqrt{u})(3+\sqrt{u})\)

Try it.

\((9-\sqrt{w})(2+\sqrt{w})\)

Solution

\(18+7\sqrt{w}-w\)

Try it.

\((7+2\sqrt{m})(4+9\sqrt{m})\)

Try it.

\((6+5\sqrt{n})(11+3\sqrt{n})\)

Solution

\(66+73\sqrt{n}+15n\)

Try it.

ⓐ \({(3+\sqrt{5})}^{2}\)
ⓑ \({(2-5\sqrt{3})}^{2}\)

Try it.

ⓐ \({(4+\sqrt{11})}^{2}\) ⓑ \({(3-2\sqrt{5})}^{2}\)

Solution

ⓐ \(27+8\sqrt{11}\) ⓑ \(29-12\sqrt{5}\)

Try it.

ⓐ \({(9-\sqrt{6})}^{2}\) ⓑ \({(10+3\sqrt{7})}^{2}\)

Try it.

ⓐ \({(5-\sqrt{10})}^{2}\) ⓑ \({(8+3\sqrt{2})}^{2}\)

Solution

ⓐ \(35-10\sqrt{10}\) ⓑ \(82+48\sqrt{2}\)

Try it.

\((3-\sqrt{5})(3+\sqrt{5})\)

Try it.

\((10-\sqrt{3})(10+\sqrt{3})\)

Solution

Try it.

\((4+\sqrt{2})(4-\sqrt{2})\)

Try it.

\((7+\sqrt{10})(7-\sqrt{10})\)

Solution

Try it.

\((4+9\sqrt{3})(4-9\sqrt{3})\)

Try it.

\((1+8\sqrt{2})(1-8\sqrt{2})\)

Solution

\(-127\)

Try it.

\((12-5\sqrt{5})(12+5\sqrt{5})\)

Try it.

\((9-4\sqrt{3})(9+4\sqrt{3})\)

Solution

Mixed Practice

In the following exercises, simplify.

Try it.

\(\sqrt{3}\cdot \sqrt{21}\)

Try it.

\((4\sqrt{6})(\text{-}\sqrt{18})\)

Solution

\(-24\sqrt{3}\)

Try it.

\((-5+\sqrt{7})(6+\sqrt{21})\)

Try it.

\((-5\sqrt{7})(6\sqrt{21})\)

Solution

\(-210\sqrt{3}\)

Try it.

\((-4\sqrt{2})(2\sqrt{18})\)

Try it.

\((\sqrt{35{y}^{3}})(\sqrt{7{y}^{3}})\)

Solution

\(7{y}^{3}\sqrt{5}\)

Try it.

\((4\sqrt{12{x}^{5}})(2\sqrt{6{x}^{3}})\)

Try it.

\({(\sqrt{29})}^{2}\)

Solution

\(29\)

Try it.

\((-4\sqrt{17})(-3\sqrt{17})\)

Try it.

\((-4+\sqrt{17})(-3+\sqrt{17})\)

Solution

\(29-7\sqrt{17}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

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Practica (40)

Încearcă fiecare pe hârtie mai întâi. Revelează răspunsul la verificare; cele verificate pot fi deschise în soluționare pentru fiecare pas.

Simplify: \((3u)(8v)\).

Revelează răspunsul

\(24uv\)
Simplify: \(6(12-7n)\).

Revelează răspunsul

\(72-42n\)
Simplify: \((2+a)(4-a)\).

Revelează răspunsul

\(8+2a+{a}^{2}\)
Simplify: ⓐ \(\sqrt{2}\cdot \sqrt{6}\) ⓑ \((4\sqrt{3})(2\sqrt{12})\).

Revelează răspunsul

ⓐ
\(\sqrt{2}\cdot \sqrt{6}\)
Multiply using the Product Property. \(\sqrt{12}\)
Simplify the radical. \(\sqrt{4}\cdot \sqrt{3}\)
Simplify. \(2\sqrt{3}\)
ⓑ
\((4\sqrt{3})(2\sqrt{12})\)
Multiply using the Product Property. \(8\sqrt{36}\)
Simplify the radical. \(8\cdot 6\)
Simplify. \(48\)
Notice that in (b) we multiplied the coefficients and multiplied the radicals. Also, we did not simplify \(\sqrt{12}\). We waited to get the product and then simplified.
Simplify: ⓐ \(\sqrt{3}\cdot \sqrt{6}\) ⓑ \((2\sqrt{6})(3\sqrt{12})\).

Revelează răspunsul

ⓐ \(3\sqrt{2}\) ⓑ \(36\sqrt{2}\)
Simplify: ⓐ \(\sqrt{5}\cdot \sqrt{10}\) ⓑ \((6\sqrt{3})(5\sqrt{6})\).

Revelează răspunsul

ⓐ \(5\sqrt{2}\) ⓑ \(90\sqrt{2}\)
Simplify: \((6\sqrt{2})(3\sqrt{10})\).

Revelează răspunsul

\((6\sqrt{2})(3\sqrt{10})\)
Multiply using the Product Property. \(18\sqrt{20}\)
Simplify the radical. \(18\sqrt{4}\cdot \sqrt{5}\)
Simplify. \(18\cdot 2\cdot \sqrt{5}\)
\(36\sqrt{5}\)
Simplify: \((3\sqrt{2})(2\sqrt{30})\).

Revelează răspunsul

\(12\sqrt{15}\)
Simplify: \((3\sqrt{3})(3\sqrt{6})\).

Revelează răspunsul

\(27\sqrt{2}\)
Simplify: ⓐ \((\sqrt{8{x}^{3}})(\sqrt{3x})\) ⓑ \((\sqrt{20{y}^{2}})(\sqrt{5{y}^{3}})\).

Revelează răspunsul

ⓐ

\((\sqrt{8{x}^{3}})(\sqrt{3x})\)
Multiply using the Product Property. \(\sqrt{24{x}^{4}}\)
Simplify the radical. \(\sqrt{4{x}^{4}}\cdot \sqrt{6}\)
Simplify. \(2{x}^{2}\sqrt{6}\)
ⓑ

\((\sqrt{20{y}^{2}})(\sqrt{5{y}^{3}})\)
Multiply using the Product Property. \(\sqrt{100{y}^{5}}\)
Simplify the radical. \(10{y}^{2}\sqrt{y}\)
Simplify: ⓐ \((\sqrt{6{x}^{3}})(\sqrt{3x})\) ⓑ \((\sqrt{2{y}^{3}})(\sqrt{50{y}^{2}})\).

Revelează răspunsul

ⓐ \(3{x}^{2}\sqrt{2}\) ⓑ \(10{y}^{2}\sqrt{y}\)
Simplify: ⓐ \((\sqrt{6{x}^{5}})(\sqrt{2x})\) ⓑ \((\sqrt{12{y}^{2}})(\sqrt{3{y}^{5}})\).

Revelează răspunsul

ⓐ \(2{x}^{3}\sqrt{3}\) ⓑ \(6{y}^{3}\sqrt{y}\)
Simplify: \((10\sqrt{6{p}^{3}})(3\sqrt{18p})\).

Revelează răspunsul

\((10\sqrt{6{p}^{3}})(3\sqrt{18p})\)
Multiply. \(30\sqrt{108{p}^{4}}\)
Simplify the radical. \(30\sqrt{36{p}^{4}}\cdot \sqrt{3}\)
\(30\cdot 6{p}^{2}\cdot \sqrt{3}\)
\(180{p}^{2}\sqrt{3}\)
Simplify: \((6\sqrt{2{x}^{2}})(8\sqrt{45{x}^{4}})\).

Revelează răspunsul

\(144{x}^{3}\sqrt{10}\)
Simplify: \((2\sqrt{6{y}^{4}})(12\sqrt{30y})\).

Revelează răspunsul

\(144{y}^{2}\sqrt{5y}\)
Simplify: ⓐ \({(\sqrt{2})}^{2}\) ⓑ \({(\text{-}\sqrt{11})}^{2}\).

Revelează răspunsul

ⓐ

\({(\sqrt{2})}^{2}\)
Rewrite as a product. \((\sqrt{2})(\sqrt{2})\)
Multiply. \(\sqrt{4}\)
Simplify. \(2\)
ⓑ

\({(\text{-}\sqrt{11})}^{2}\)
Rewrite as a product. \((\text{-}\sqrt{11})(\text{-}\sqrt{11})\)
Multiply. \(\sqrt{121}\)
Simplify. \(11\)
Simplify: ⓐ \({(\sqrt{12})}^{2}\) ⓑ \({(\text{-}\sqrt{15})}^{2}\).

Revelează răspunsul

ⓐ 12 ⓑ \(15\)
Simplify: ⓐ \({(\sqrt{16})}^{2}\) ⓑ \({(\text{-}\sqrt{20})}^{2}\).

Revelează răspunsul

ⓐ 16 ⓑ 20
Simplify: ⓐ \((2\sqrt{3})(8\sqrt{3})\) ⓑ \({(3\sqrt{6})}^{2}\).

Revelează răspunsul

ⓐ

\((2\sqrt{3})(8\sqrt{3})\)
Multiply. Remember, \({(\sqrt{3})}^{2}=3.\) \(16\cdot 3\)
Simplify. \(48\)
ⓑ

\({(3\sqrt{6})}^{2}\)
Multiply. \(9\cdot 6\)
Simplify. \(54\)
Simplify: ⓐ \((6\sqrt{11})(5\sqrt{11})\) ⓑ \({(5\sqrt{8})}^{2}\).

Revelează răspunsul

ⓐ 330 ⓑ 200
Simplify: ⓐ \((3\sqrt{7})(10\sqrt{7})\) ⓑ \({(-4\sqrt{6})}^{2}\).

Revelează răspunsul

ⓐ 210 ⓑ 96
Simplify: ⓐ \(3(5-\sqrt{2})\) ⓑ \(\sqrt{2}(4-\sqrt{10})\).

Revelează răspunsul

ⓐ

\(3(5-\sqrt{2})\)
Distribute. \(15-3\sqrt{2}\)
ⓑ

\(\sqrt{2}(4-\sqrt{10})\)
Distribute. \(4\sqrt{2}-\sqrt{20}\)
\(4\sqrt{2}-2\sqrt{5}\)
Simplify: ⓐ \(2(3-\sqrt{5})\) ⓑ \(\sqrt{3}(2-\sqrt{18})\).

Revelează răspunsul

ⓐ \(6-2\sqrt{5}\) ⓑ \(2\sqrt{3}-3\sqrt{6}\)
Simplify: ⓐ \(6(2+\sqrt{6})\) ⓑ \(\sqrt{7}(1+\sqrt{14})\).

Revelează răspunsul

ⓐ \(12+6\sqrt{6}\) ⓑ \(\sqrt{7}+7\sqrt{2}\)
Simplify: ⓐ \(\sqrt{5}(7+2\sqrt{5})\) ⓑ \(\sqrt{6}(\sqrt{2}+\sqrt{18})\).

Revelează răspunsul

ⓐ

\(\sqrt{5}(7+2\sqrt{5})\)
Multiply. \(7\sqrt{5}+2\cdot 5\)
Simplify. \(7\sqrt{5}+10\)
\(10+7\sqrt{5}\)
ⓑ

\(\sqrt{6}(\sqrt{2}+\sqrt{18})\)
Multiply. \(\sqrt{12}+\sqrt{108}\)
Simplify. \(\sqrt{4}\cdot \sqrt{3}+\sqrt{36}\cdot \sqrt{3}\)
\(2\sqrt{3}+6\sqrt{3}\)
Combine like radicals. \(8\sqrt{3}\)
Simplify: ⓐ \(\sqrt{6}(1+3\sqrt{6})\) ⓑ \(\sqrt{12}(\sqrt{3}+\sqrt{24})\).

Revelează răspunsul

ⓐ \(18+\sqrt{6}\) ⓑ \(6+12\sqrt{2}\)
Simplify: ⓐ \(\sqrt{8}(2-5\sqrt{8})\) ⓑ \(\sqrt{14}(\sqrt{2}+\sqrt{42})\).

Revelează răspunsul

ⓐ \(-40+4\sqrt{2}\) ⓑ \(2\sqrt{7}+14\sqrt{3}\)
Simplify: \((2+\sqrt{3})(4-\sqrt{3})\).

Revelează răspunsul

\((2+\sqrt{3})(4-\sqrt{3})\)
Multiply. \(8-2\sqrt{3}+4\sqrt{3}-3\)
Combine like terms. \(5+2\sqrt{3}\)
Simplify: \((1+\sqrt{6})(3-\sqrt{6})\).

Revelează răspunsul

\(-3+2\sqrt{6}\)
Simplify: \((4-\sqrt{10})(2+\sqrt{10})\).

Revelează răspunsul

\(-2+2\sqrt{10}\)
Simplify: \((3-2\sqrt{7})(4-2\sqrt{7})\).

Revelează răspunsul

\((3-2\sqrt{7})(4-2\sqrt{7})\)
Multiply. \(12-6\sqrt{7}-8\sqrt{7}+4\cdot 7\)
Simplify. \(12-6\sqrt{7}-8\sqrt{7}+28\)
Combine like terms. \(40-14\sqrt{7}\)
Simplify: \((6-3\sqrt{7})(3+4\sqrt{7})\).

Revelează răspunsul

\(-66+15\sqrt{7}\)
Simplify: \((2-3\sqrt{11})(4-\sqrt{11})\).

Revelează răspunsul

\(41-14\sqrt{11}\)
Simplify: \((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\).

Revelează răspunsul

\((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
Multiply. \(3\cdot 2+12\sqrt{10}-\sqrt{10}-4\cdot 5\)
Simplify. \(6+12\sqrt{10}-\sqrt{10}-20\)
Combine like terms. \(-14+11\sqrt{10}\)
Simplify: \((5\sqrt{3}-\sqrt{7})(\sqrt{3}+2\sqrt{7})\).

Revelează răspunsul

\(1+9\sqrt{21}\)
Simplify: \((\sqrt{6}-3\sqrt{8})(2\sqrt{6}+\sqrt{8})\)

Revelează răspunsul

\(-12-20\sqrt{3}\)
Simplify: \((4-2\sqrt{x})(1+3\sqrt{x})\).

Revelează răspunsul

\((4-2\sqrt{x})(1+3\sqrt{x})\)
Multiply. \(4+12\sqrt{x}-2\sqrt{x}-6x\)
Combine like terms. \(4+10\sqrt{x}-6x\)
Simplify: \((6-5\sqrt{m})(2+3\sqrt{m})\).

Revelează răspunsul

\(12+8\sqrt{m}-15m\)
Simplify: \((10+3\sqrt{n})(1-5\sqrt{n})\).

Revelează răspunsul

\(10-47\sqrt{n}-15n\)
Simplify: ⓐ \({(2+\sqrt{3})}^{2}\) ⓑ \({(4-2\sqrt{5})}^{2}\).
Revelează răspunsul
Be sure to include the \(2ab\) term when squaring a binomial.
1. ⓐ
  
  Multiply using the binomial square pattern.
  Simplify.
  Combine like terms.
2. ⓑ
  
  Multiply using the binomial square pattern.
  Simplify.
  Combine like terms.

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Simbole utilizate aici

Atinge orice simbol pentru definiţia completă, o imagine şi ce înseamnă fiecare literă.

Cum să: Multiply Square Roots

Multiply square roots
Use polynomial multiplication to multiply square roots
The FOIL method can be used to multiply binomials containing radicals.

Întrebări pe care oamenii le întreabă

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Partea acestei pagini este adaptată de la OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensată şi reexplicată aici; erorile sunt ale noastre. Pentru că sursa sa este licențiată pentru utilizare necomercială, această lecție este gratuită, nu vinde nimic și este împărtășită sub aceeași licență.

Mai multe în Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

	\((10\sqrt{6{p}^{3}})(3\sqrt{18p})\)
Multiply.	\(30\sqrt{108{p}^{4}}\)
Simplify the radical.	\(30\sqrt{36{p}^{4}}\cdot \sqrt{3}\)
	\(30\cdot 6{p}^{2}\cdot \sqrt{3}\)
	\(180{p}^{2}\sqrt{3}\)

	\({(\sqrt{2})}^{2}\)
Rewrite as a product.	\((\sqrt{2})(\sqrt{2})\)
Multiply.	\(\sqrt{4}\)
Simplify.	\(2\)

	\({(\text{-}\sqrt{11})}^{2}\)
Rewrite as a product.	\((\text{-}\sqrt{11})(\text{-}\sqrt{11})\)
Multiply.	\(\sqrt{121}\)
Simplify.	\(11\)

	\((2\sqrt{3})(8\sqrt{3})\)
Multiply. Remember, \({(\sqrt{3})}^{2}=3.\)	\(16\cdot 3\)
Simplify.	\(48\)


Multiply using the binomial square pattern.
Simplify.
Combine like terms.

	\({(3\sqrt{6})}^{2}\)
Multiply.	\(9\cdot 6\)
Simplify.	\(54\)