maths.free › Algebra › Roots and Radicals › Multiply Square Roots

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.

Most te. Nincs számológép rendezi ezt, de a darabjai komposztálhatóak. Próbálja ki az egyiket lentebb, vagy gépelje be a sajátját.

Gyakorlat (40)

Próbálja ki mindet papíron először. Fedezd fel a választ, hogy ellenőrizze; ellenőrzött is lehet nyitni a megoldás minden lépésnél.

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

\((6\sqrt{2})(3\sqrt{10})\)
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: \((3\sqrt{2})(2\sqrt{30})\).

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

ⓐ

\((\sqrt{8{x}^{3}})(\sqrt{3x})\)
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: ⓐ \((\sqrt{6{x}^{3}})(\sqrt{3x})\) ⓑ \((\sqrt{2{y}^{3}})(\sqrt{50{y}^{2}})\).

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

\((10\sqrt{6{p}^{3}})(3\sqrt{18p})\)
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: \((6\sqrt{2{x}^{2}})(8\sqrt{45{x}^{4}})\).

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

ⓐ

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

ⓐ

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

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

ⓐ

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

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

ⓐ

\(\sqrt{5}(7+2\sqrt{5})\)
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: ⓐ \(\sqrt{6}(1+3\sqrt{6})\) ⓑ \(\sqrt{12}(\sqrt{3}+\sqrt{24})\).

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

\((3-2\sqrt{7})(4-2\sqrt{7})\)
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: \((6-3\sqrt{7})(3+4\sqrt{7})\).

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

\((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
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: \((5\sqrt{3}-\sqrt{7})(\sqrt{3}+2\sqrt{7})\).

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

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

Mutasd meg a választ!

\(10-47\sqrt{n}-15n\)
Simplify: ⓐ \({(2+\sqrt{3})}^{2}\) ⓑ \({(4-2\sqrt{5})}^{2}\).
Mutasd meg a választ!
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.

Dolgozz tovább!

Egy ingyenes fiók jegyzeteket ad minden leckéről, egy feljegyzést arról, hogy mit végeztél el, a megoldott problémáidról egy helyen, és egy oktatót, akit megkérdezhetsz erről az oldalról. A matematika maga mindenki számára nyitva áll, bejelentkezve vagy sem.

Regisztrálj! Bejelentkezés

Itt használt szimbólumok

Koppintson a teljes definícióra, a képre és arra, hogy mit jelent minden betű.

Hogyan kell: Multiply Square Roots

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

Kérdések, amiket az emberek feltesznek

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.

Az oldal részei adaptálva OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Össze van zavarodva és újra megmagyarázva; a hibák a miénk. Mivel a forrás nem kereskedelmi felhasználásra engedélyezett, ez a lecke ingyenes, nem ad el semmit, és ugyanazon licenc alatt osztozik.

Még több 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.	\(54\)