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.

Ҳоло шумо Ҳеҷ ҳисобкунак инро ҳал намекунад, лекин қисмҳои он ҳисобшавандаанд. Яке аз инҳоро дар поён санҷед ё худи худро ворид кунед.

Машқ (40)

Ҳар як саволро аввал дар коғаз санҷед. Ҷавобро барои санҷиш нишон диҳед; ҷавобҳои санҷидашударо дар ҳалкунанда барои ҳар қадам кушода метавонед.

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

    Ҷавоби ҷавобро нишон диҳед

    \(24uv\)

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

    Ҷавоби ҷавобро нишон диҳед

    \(72-42n\)

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ \(3\sqrt{2}\) ⓑ \(36\sqrt{2}\)

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ \(5\sqrt{2}\) ⓑ \(90\sqrt{2}\)

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

    Ҷавоби ҷавобро нишон диҳед
    \((6\sqrt{2})(3\sqrt{10})\)
    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: \((3\sqrt{2})(2\sqrt{30})\).

    Ҷавоби ҷавобро нишон диҳед

    \(12\sqrt{15}\)

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

    Ҷавоби ҷавобро нишон диҳед

    \(27\sqrt{2}\)

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ 12 ⓑ \(15\)

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ 16 ⓑ 20

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ

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

    ⓑ

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

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ 330 ⓑ 200

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ 210 ⓑ 96

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ

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

    ⓑ

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

    ⓐ

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед
    \((3-2\sqrt{7})(4-2\sqrt{7})\)
    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: \((6-3\sqrt{7})(3+4\sqrt{7})\).

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед
    \((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
    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: \((5\sqrt{3}-\sqrt{7})(\sqrt{3}+2\sqrt{7})\).

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

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

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

    Ҷавоби ҷавобро нишон диҳед

    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.

Кор кардан

Дар ҳисоби ройгон ба ҳар як дарс қайдҳо илова карда мешаванд, қайди он, ки шумо чӣ кор кардаед, масъалаҳои ҳалшуда дар як ҷо ва муаллиме, ки шумо метавонед дар бораи ин саҳифа пурсед. Математика барои ҳама кушода аст, хоҳ ворид шуда бошад ё на.

Бақайдгирӣ Ворид шудан

Символы, используемые здесь

Барои муайянкунии пурраи маъно, тасвир ва маънои ҳар як ҳарф дар он, ба ҳар як аломат пахш кунед.

Чӣ тавр: 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.

Саволҳои маъмул

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.

Қисми ин саҳифа аз OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Дар ин ҷо барои мо чизе нест, ки мо ба он бирасем. Азбаски манбаъ барои истифодаи ғайрикоммерсиалӣ иҷозатнома дода шудааст, ин дарс ройгон аст, чизе намефурӯшад ва дар асоси иҷозатномаи якхела тақсим карда мешавад.

& Иловаи забон Algebra

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