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

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.

今度はあなたの番 計算機では解けませんが、計算可能です。下の一つを試してみましょう。もしくは自分で入力してください。

練習 (40)

まず紙で解いてみてください。答えを表示して確認できます。検証済みの問題は、ソルバーで全ステップを開けます。

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

    答えを表示

    \(24uv\)

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

    答えを表示

    \(72-42n\)

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

    答えを表示

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

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

    答えを表示

    ⓐ

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

    答えを表示

    ⓐ \(3\sqrt{2}\) ⓑ \(36\sqrt{2}\)

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

    答えを表示

    ⓐ \(5\sqrt{2}\) ⓑ \(90\sqrt{2}\)

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

    答えを表示
    \((6\sqrt{2})(3\sqrt{10})\)
    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: \((3\sqrt{2})(2\sqrt{30})\).

    答えを表示

    \(12\sqrt{15}\)

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

    答えを表示

    \(27\sqrt{2}\)

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

    答えを表示

    ⓐ

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

    答えを表示

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

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

    答えを表示

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

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

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

    答えを表示

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

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

    答えを表示

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

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

    答えを表示

    ⓐ

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

    答えを表示

    ⓐ 12 ⓑ \(15\)

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

    答えを表示

    ⓐ 16 ⓑ 20

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

    答えを表示

    ⓐ

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

    ⓑ

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

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

    答えを表示

    ⓐ 330 ⓑ 200

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

    答えを表示

    ⓐ 210 ⓑ 96

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

    答えを表示

    ⓐ

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

    ⓑ

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

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

    答えを表示

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

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

    答えを表示

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

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

    答えを表示

    ⓐ

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

    答えを表示

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

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

    答えを表示

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

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

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

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

    答えを表示

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

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

    答えを表示

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

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

    答えを表示
    \((3-2\sqrt{7})(4-2\sqrt{7})\)
    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: \((6-3\sqrt{7})(3+4\sqrt{7})\).

    答えを表示

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

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

    答えを表示

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

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

    答えを表示
    \((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
    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: \((5\sqrt{3}-\sqrt{7})(\sqrt{3}+2\sqrt{7})\).

    答えを表示

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

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

    答えを表示

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

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

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

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

    答えを表示

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

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

    答えを表示

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

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

    答えを表示

    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.

自分の解答を記録しよう

無料アカウントでは、すべてのレッスンにノートを追加し、完成したことの記録、解いた問題を一つの場所に保存し、このページについて質問できる先生を追加します。数学自体はログインしたかどうかに関係なく誰でも利用できます。

登録 ログイン

ここで使用された記号

どの記号をタップしても、定義、画像、それぞれの文字の意味が表示されます。

どうやって: 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.

よくある質問

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.

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). ここでは要約して説明し直しています。誤りは私たちの責任です。 ソースは非商用利用のためにライセンスされているため、このレッスンは無料で、売ることはなく、同じライセンスの下で共有されている。

More in Algebra

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