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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.

Sekarang kau Tiada kalkulator yang dapat menyelesaikan ini, tetapi bahagian-bahagian boleh dikira. Cuba salah satu di bawah, atau taipkan sendiri.

Latihan (40)

Cuba setiap satu di atas kertas dahulu. Tunjukkan jawapan untuk semak; yang disahkan boleh dibuka dalam penyelesai bagi setiap langkah.

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

    Tunjukkan jawapan

    \(24uv\)

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

    Tunjukkan jawapan

    \(72-42n\)

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

    ⓐ

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

    Tunjukkan jawapan

    ⓐ \(3\sqrt{2}\) ⓑ \(36\sqrt{2}\)

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

    Tunjukkan jawapan

    ⓐ \(5\sqrt{2}\) ⓑ \(90\sqrt{2}\)

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

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

    Tunjukkan jawapan

    \(12\sqrt{15}\)

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

    Tunjukkan jawapan

    \(27\sqrt{2}\)

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

    Tunjukkan jawapan

    ⓐ

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

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

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

    ⓐ

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

    Tunjukkan jawapan

    ⓐ 12 ⓑ \(15\)

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

    Tunjukkan jawapan

    ⓐ 16 ⓑ 20

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

    Tunjukkan jawapan

    ⓐ

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

    ⓑ

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

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

    Tunjukkan jawapan

    ⓐ 330 ⓑ 200

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

    Tunjukkan jawapan

    ⓐ 210 ⓑ 96

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

    Tunjukkan jawapan

    ⓐ

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

    ⓑ

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

    ⓐ

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

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

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

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

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

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

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

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

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

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

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

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

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

    Tunjukkan jawapan

    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.

Jaga kerja awak sendiri.

Akaun percuma menambah nota pada setiap pengajaran, rekod apa yang anda selesaikan, masalah yang anda selesaikan di satu tempat, dan seorang guru yang anda boleh tanya tentang halaman ini. Matematik itu sendiri terbuka kepada semua orang, sama ada anda log masuk atau tidak.

Daftar Masuk Log masuk

Simbol yang digunakan di sini

Tekan simbol mana-mana untuk definisi penuh, gambar, dan apa makna setiap huruf dalamnya.

Bagaimana untuk: 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.

Soalan yang orang tanya

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.

Bahagian halaman ini diadaptasi dari OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Dikemas kini dan dijelaskan di sini; kesilapan adalah kami. Kerana sumbernya dilesenkan untuk penggunaan bukan komersial, pengajaran ini adalah percuma, tidak menjual apa-apa, dan dikongsi di bawah lesen yang sama.

Lebih dalam Algebra

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