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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

97

Try it.

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

Try it.

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

Solution

39

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

33

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.

Ora tu Nessuna calcolatrice risolve questa domanda, ma le sue parti sono calcolabili. Provane una qui sotto, o digita la tua.

Pratica (40)

Provare ciascuno su carta prima. Rivelare la risposta per controllare; quelli verificati possono essere aperti nel risolutore per ogni passo.

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

    Rivela la risposta

    \(24uv\)

  2. Simplify: \(6(12-7n)\).

    Rivela la risposta

    \(72-42n\)

  3. Simplify: \((2+a)(4-a)\).

    Rivela la risposta

    \(8+2a+{a}^{2}\)

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

    Rivela la risposta

    ⓐ

    \(\sqrt{2}\cdot \sqrt{6}\)
    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.

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

    Rivela la risposta

    ⓐ \(3\sqrt{2}\) ⓑ \(36\sqrt{2}\)

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

    Rivela la risposta

    ⓐ \(5\sqrt{2}\) ⓑ \(90\sqrt{2}\)

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

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

  8. Simplify: \((3\sqrt{2})(2\sqrt{30})\).

    Rivela la risposta

    \(12\sqrt{15}\)

  9. Simplify: \((3\sqrt{3})(3\sqrt{6})\).

    Rivela la risposta

    \(27\sqrt{2}\)

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

    Rivela la risposta

    ⓐ

    \((\sqrt{8{x}^{3}})(\sqrt{3x})\)
    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.\(10{y}^{2}\sqrt{y}\)

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

    Rivela la risposta

    ⓐ \(3{x}^{2}\sqrt{2}\) ⓑ \(10{y}^{2}\sqrt{y}\)

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

    Rivela la risposta

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

  13. Simplify: \((10\sqrt{6{p}^{3}})(3\sqrt{18p})\).

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

  14. Simplify: \((6\sqrt{2{x}^{2}})(8\sqrt{45{x}^{4}})\).

    Rivela la risposta

    \(144{x}^{3}\sqrt{10}\)

  15. Simplify: \((2\sqrt{6{y}^{4}})(12\sqrt{30y})\).

    Rivela la risposta

    \(144{y}^{2}\sqrt{5y}\)

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

    Rivela la risposta

    ⓐ

    \({(\sqrt{2})}^{2}\)
    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.\(11\)

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

    Rivela la risposta

    ⓐ 12 ⓑ \(15\)

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

    Rivela la risposta

    ⓐ 16 ⓑ 20

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

    Rivela la risposta

    ⓐ

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

    ⓑ

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

  20. Simplify: ⓐ \((6\sqrt{11})(5\sqrt{11})\) ⓑ \({(5\sqrt{8})}^{2}\).

    Rivela la risposta

    ⓐ 330 ⓑ 200

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

    Rivela la risposta

    ⓐ 210 ⓑ 96

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

    Rivela la risposta

    ⓐ

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

    ⓑ

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

  23. Simplify: ⓐ \(2(3-\sqrt{5})\) ⓑ \(\sqrt{3}(2-\sqrt{18})\).

    Rivela la risposta

    ⓐ \(6-2\sqrt{5}\) ⓑ \(2\sqrt{3}-3\sqrt{6}\)

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

    Rivela la risposta

    ⓐ \(12+6\sqrt{6}\) ⓑ \(\sqrt{7}+7\sqrt{2}\)

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

    Rivela la risposta

    ⓐ

    \(\sqrt{5}(7+2\sqrt{5})\)
    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.\(8\sqrt{3}\)

  26. Simplify: ⓐ \(\sqrt{6}(1+3\sqrt{6})\) ⓑ \(\sqrt{12}(\sqrt{3}+\sqrt{24})\).

    Rivela la risposta

    ⓐ \(18+\sqrt{6}\) ⓑ \(6+12\sqrt{2}\)

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

    Rivela la risposta

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

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

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

  29. Simplify: \((1+\sqrt{6})(3-\sqrt{6})\).

    Rivela la risposta

    \(-3+2\sqrt{6}\)

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

    Rivela la risposta

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

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

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

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

    Rivela la risposta

    \(-66+15\sqrt{7}\)

  33. Simplify: \((2-3\sqrt{11})(4-\sqrt{11})\).

    Rivela la risposta

    \(41-14\sqrt{11}\)

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

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

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

    Rivela la risposta

    \(1+9\sqrt{21}\)

  36. Simplify: \((\sqrt{6}-3\sqrt{8})(2\sqrt{6}+\sqrt{8})\)

    Rivela la risposta

    \(-12-20\sqrt{3}\)

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

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

  38. Simplify: \((6-5\sqrt{m})(2+3\sqrt{m})\).

    Rivela la risposta

    \(12+8\sqrt{m}-15m\)

  39. Simplify: \((10+3\sqrt{n})(1-5\sqrt{n})\).

    Rivela la risposta

    \(10-47\sqrt{n}-15n\)

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

    Rivela la risposta

    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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Simboli utilizzati qui

Toccare qualsiasi simbolo per la definizione completa, un'immagine, e che cosa ogni lettera in esso significa.

Come: Multiply Square Roots

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

Domande che la gente chiede

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.

Parti di questa pagina sono adattate da OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensato e ri-spiegato qui; gli errori sono nostri. Poiché la sua fonte è autorizzata per uso non commerciale, questa lezione è gratuita, non vende nulla e viene condivisa con la stessa licenza.

Altro in Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value