Simplify Square Roots

Use the Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that \({(ab)}^{m}={a}^{m}{b}^{m}\). The corresponding property of square roots says that \(\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}\).

We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in .

How To Use the Product Property to Simplify a Square Root

Try it.

Simplify: \(\sqrt{50}\).

Solution

Notice in the previous example that the simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\), which is the product of an integer and a square root. We always write the integer in front of the square root.

Example

Try it.

Simplify: \(\sqrt{500}\).

Solution

	\(\sqrt{500}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{100\cdot 5}\)
Rewrite the radical as the product of two radicals.	\(\sqrt{100}\cdot \sqrt{5}\)
Simplify.	\(10\sqrt{5}\)

We could use the simplified form \(10\sqrt{5}\) to estimate \(\sqrt{500}\). We know 5 is between 2 and 3, and \(\sqrt{500}\) is \(10\sqrt{5}\). So \(\sqrt{500}\) is between 20 and 30.

The next example is much like the previous examples, but with variables.

Example

Try it.

Simplify: \(\sqrt{{x}^{3}}\).

Solution

	\(\sqrt{{x}^{3}}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{{x}^{2}\cdot x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt{{x}^{2}}\cdot \sqrt{x}\)
Simplify.	\(x\sqrt{x}\)

We follow the same procedure when there is a coefficient in the radical, too.

Example

Try it.

Simplify: \(\sqrt{25{y}^{5}.}\)

Solution

	\(\sqrt{25{y}^{5}}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{25{y}^{4}\cdot y}\)
Rewrite the radical as the product of two radicals.	\(\sqrt{25{y}^{4}}\cdot \sqrt{y}\)
Simplify.	\(5{y}^{2}\sqrt{y}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

Example

Try it.

Simplify: \(\sqrt{\frac{9}{64}}\).

Solution

\(\begin{array}{llll} & & & \ \sqrt{\frac{9}{64}} \\ \text{Since}\ {(\frac{3}{8})}^{2}=\frac{9}{64} & & & \ \frac{3}{8}\end{array}\)

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

Example

Try it.

Simplify: \(\sqrt{\frac{45}{80}}\).

Solution

	\(\sqrt{\frac{45}{80}}\)
Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.	\(\sqrt{\frac{5\cdot 9}{5\cdot 16}}\)
Simplify the fraction by removing common factors.	\(\sqrt{\frac{9}{16}}\)
\(\text{Simplify.}\ {(\frac{3}{4})}^{2}=\frac{9}{16}\)	\(\frac{3}{4}\)

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},a\ne 0\).

Example

Try it.

Simplify: \(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\).

Solution

	\(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\)
Simplify the fraction inside the radical first.
Divide the like bases by subtracting the exponents.	\(\sqrt{{m}^{2}}\)
Simplify.	\(m\)

Example

Try it.

Simplify: \(\sqrt{\frac{48{p}^{7}}{3{p}^{3}}}\).

Solution

	\(\sqrt{\frac{48{p}^{7}}{3{p}^{3}}}\)
Simplify the fraction inside the radical first.	\(\sqrt{16{p}^{4}}\)
Simplify.	\(4{p}^{2}\)

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

\[{(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne 0\]

We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

• Simplified Square Root \(\sqrt{a}\) is considered simplified if \(a\) has no perfect-square factors.
• Product Property of Square Roots If a, b are non-negative real numbers, then
  \[\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}\]
• Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
  1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
  2. Use the product rule to rewrite the radical as the product of two radicals.
  3. Simplify the square root of the perfect square.
• Quotient Property of Square Roots If a, b are non-negative real numbers and \(b\ne 0\), then
  \[\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\]
• Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
  1. Simplify the fraction in the radicand, if possible.
  2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
  3. Simplify the radicals in the numerator and the denominator.

Simplify Square Roots

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

Try it.

\(\sqrt{27}\)

Solution

\(3\sqrt{3}\)

Try it.

\(\sqrt{80}\)

Try it.

\(\sqrt{125}\)

Solution

\(5\sqrt{5}\)

Try it.

\(\sqrt{96}\)

Try it.

\(\sqrt{200}\)

Solution

\(10\sqrt{2}\)

Try it.

\(\sqrt{147}\)

Try it.

\(\sqrt{450}\)

Solution

\(15\sqrt{2}\)

Try it.

\(\sqrt{252}\)

Try it.

\(\sqrt{800}\)

Solution

\(20\sqrt{2}\)

Try it.

\(\sqrt{288}\)

Try it.

\(\sqrt{675}\)

Solution

\(15\sqrt{3}\)

Try it.

\(\sqrt{1250}\)

Try it.

\(\sqrt{{x}^{7}}\)

Solution

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

Try it.

\(\sqrt{{y}^{11}}\)

Try it.

\(\sqrt{{p}^{3}}\)

Solution

\(p\sqrt{p}\)

Try it.

\(\sqrt{{q}^{5}}\)

Try it.

\(\sqrt{{m}^{13}}\)

Solution

\({m}^{6}\sqrt{m}\)

Try it.

\(\sqrt{{n}^{21}}\)

Try it.

\(\sqrt{{r}^{25}}\)

Solution

\({r}^{12}\sqrt{r}\)

Try it.

\(\sqrt{{s}^{33}}\)

Try it.

\(\sqrt{49{n}^{17}}\)

Solution

\(7{n}^{8}\sqrt{n}\)

Try it.

\(\sqrt{25{m}^{9}}\)

Try it.

\(\sqrt{81{r}^{15}}\)

Solution

\(9{r}^{7}\sqrt{r}\)

Try it.

\(\sqrt{100{s}^{19}}\)

Try it.

\(\sqrt{98{m}^{5}}\)

Solution

\(7{m}^{2}\sqrt{2m}\)

Try it.

\(\sqrt{32{n}^{11}}\)

Try it.

\(\sqrt{125{r}^{13}}\)

Solution

\(5{r}^{6}\sqrt{5r}\)

Try it.

\(\sqrt{80{s}^{15}}\)

Try it.

\(\sqrt{200{p}^{13}}\)

Solution

\(10{p}^{6}\sqrt{2p}\)

Try it.

\(\sqrt{128{q}^{3}}\)

Try it.

\(\sqrt{242{m}^{23}}\)

Solution

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

Try it.

\(\sqrt{175{n}^{13}}\)

Try it.

\(\sqrt{147{m}^{7}{n}^{11}}\)

Solution

\(7{m}^{3}{n}^{5}\sqrt{3mn}\)

Try it.

\(\sqrt{48{m}^{7}{n}^{5}}\)

Try it.

\(\sqrt{75{r}^{13}{s}^{9}}\)

Solution

\(5{r}^{6}{s}^{4}\sqrt{3rs}\)

Try it.

\(\sqrt{96{r}^{3}{s}^{3}}\)

Try it.

\(\sqrt{300{p}^{9}{q}^{11}}\)

Solution

\(10{p}^{4}{q}^{5}\sqrt{3pq}\)

Try it.

\(\sqrt{192{q}^{3}{r}^{7}}\)

Try it.

\(\sqrt{242{m}^{13}{n}^{21}}\)

Solution

\(11{m}^{6}{n}^{10}\sqrt{2mn}\)

Try it.

\(\sqrt{150{m}^{9}{n}^{3}}\)

Try it.

\(5+\sqrt{12}\)

Solution

\(5+2\sqrt{3}\)

Try it.

\(8+\sqrt{96}\)

Try it.

\(1+\sqrt{45}\)

Solution

\(1+3\sqrt{5}\)

Try it.

\(3+\sqrt{125}\)

Try it.

\(\frac{10-\sqrt{24}}{2}\)

Solution

\(5-\sqrt{6}\)

Try it.

\(\frac{8-\sqrt{80}}{4}\)

Try it.

\(\frac{3+\sqrt{90}}{3}\)

Solution

\(1+\sqrt{10}\)

Try it.

\(\frac{15+\sqrt{75}}{5}\)

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

Try it.

\(\sqrt{\frac{49}{64}}\)

Solution

\(\frac{7}{8}\)

Try it.

\(\sqrt{\frac{100}{36}}\)

Try it.

\(\sqrt{\frac{121}{16}}\)

Solution

\(\frac{11}{4}\)

Try it.

\(\sqrt{\frac{144}{169}}\)

Try it.

\(\sqrt{\frac{72}{98}}\)

Solution

\(\frac{6}{7}\)

Try it.

\(\sqrt{\frac{75}{12}}\)

Try it.

\(\sqrt{\frac{45}{125}}\)

Solution

\(\frac{3}{5}\)

Try it.

\(\sqrt{\frac{300}{243}}\)

Try it.

\(\sqrt{\frac{{x}^{10}}{{x}^{6}}}\)

Solution

\({x}^{2}\)

Try it.

\(\sqrt{\frac{{p}^{20}}{{p}^{10}}}\)

Try it.

\(\sqrt{\frac{{y}^{4}}{{y}^{8}}}\)

Solution

\(\frac{1}{{y}^{2}}\)

Try it.

\(\sqrt{\frac{{q}^{8}}{{q}^{14}}}\)

Try it.

\(\sqrt{\frac{200{x}^{7}}{2{x}^{3}}}\)

Solution

\(10{x}^{2}\)

Try it.

\(\sqrt{\frac{98{y}^{11}}{2{y}^{5}}}\)

Try it.

\(\sqrt{\frac{96{p}^{9}}{6p}}\)

Solution

\(4{p}^{4}\)

Try it.

\(\sqrt{\frac{108{q}^{10}}{3{q}^{2}}}\)

Try it.

\(\sqrt{\frac{36}{35}}\)

Solution

\(\frac{6}{\sqrt{35}}\)

Try it.

\(\sqrt{\frac{144}{65}}\)

Try it.

\(\sqrt{\frac{20}{81}}\)

Solution

\(\frac{2\sqrt{5}}{9}\)

Try it.

\(\sqrt{\frac{21}{196}}\)

Try it.

\(\sqrt{\frac{96{x}^{7}}{121}}\)

Solution

\(\frac{4{x}^{3}\sqrt{6x}}{11}\)

Try it.

\(\sqrt{\frac{108{y}^{4}}{49}}\)

Try it.

\(\sqrt{\frac{300{m}^{5}}{64}}\)

Solution

\(\frac{5{m}^{2}\sqrt{3m}}{4}\)

Try it.

\(\sqrt{\frac{125{n}^{7}}{169}}\)

Try it.

\(\sqrt{\frac{98{r}^{5}}{100}}\)

Solution

\(\frac{7{r}^{2}\sqrt{2r}}{10}\)

Try it.

\(\sqrt{\frac{180{s}^{10}}{144}}\)

Try it.

\(\sqrt{\frac{28{q}^{6}}{225}}\)

Solution

\(\frac{2{q}^{3}\sqrt{7}}{15}\)

Try it.

\(\sqrt{\frac{150{r}^{3}}{256}}\)

Try it.

\(\sqrt{\frac{75{r}^{9}}{{s}^{8}}}\)

Solution

\(\frac{5{r}^{4}\sqrt{3r}}{{s}^{4}}\)

Try it.

\(\sqrt{\frac{72{x}^{5}}{{y}^{6}}}\)

Try it.

\(\sqrt{\frac{28{p}^{7}}{{q}^{2}}}\)

Solution

\(\frac{2{p}^{3}\sqrt{7p}}{q}\)

Try it.

\(\sqrt{\frac{45{r}^{3}}{{s}^{10}}}\)

Try it.

\(\sqrt{\frac{100{x}^{5}}{36{x}^{3}}}\)

Solution

\(\frac{5x}{3}\)

Try it.

\(\sqrt{\frac{49{r}^{12}}{16{r}^{6}}}\)

Try it.

\(\sqrt{\frac{121{p}^{5}}{81{p}^{2}}}\)

Solution

\(\frac{11p\sqrt{p}}{9}\)

Try it.

\(\sqrt{\frac{25{r}^{8}}{64r}}\)

Try it.

\(\sqrt{\frac{32{x}^{5}{y}^{3}}{18{x}^{3}y}}\)

Solution

\(\frac{4xy}{3}\)

Try it.

\(\sqrt{\frac{75{r}^{6}{s}^{8}}{48r{s}^{4}}}\)

Try it.

\(\sqrt{\frac{27{p}^{2}q}{108{p}^{5}{q}^{3}}}\)

Solution

\(\frac{1}{2pq\sqrt{p}}\)

Try it.

\(\sqrt{\frac{50{r}^{5}{s}^{2}}{128{r}^{2}{s}^{5}}}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

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先在纸上试一个。 将答案显示为检查; 校验的答案可以在求解器中打开, 每一步都可用 。

1. Simplify: \(\frac{80}{176}\).

   通过它工作 提示

   发送答案

   \(\frac{5}{11}\)

2. Simplify: \(\frac{{n}^{9}}{{n}^{3}}\).

   通过它工作 提示

   发送答案

   \({n}^{6}\)

3. Simplify: \(\frac{{q}^{4}}{{q}^{12}}\).

   通过它工作 提示

   发送答案

   \(\frac{1}{{q}^{8}}\)

4. Simplify: \(\sqrt{50}\).

   通过它工作 提示

5. Simplify: \(\sqrt{48}\).

   通过它工作 提示

   发送答案

   \(4\sqrt{3}\)

6. Simplify: \(\sqrt{45}\).

   通过它工作 提示

   发送答案

   \(3\sqrt{5}\)

7. Simplify: \(\sqrt{500}\).

   通过它工作 提示

   发送答案

   	\(\sqrt{500}\)
   Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{100\cdot 5}\)
   Rewrite the radical as the product of two radicals.	\(\sqrt{100}\cdot \sqrt{5}\)
   Simplify.	\(10\sqrt{5}\)

8. Simplify: \(\sqrt{288}\).

   通过它工作 提示

   发送答案

   \(12\sqrt{2}\)

9. Simplify: \(\sqrt{432}\).

   通过它工作 提示

   发送答案

   \(12\sqrt{3}\)

10. Simplify: \(\sqrt{{x}^{3}}\).

    通过它工作 提示

    发送答案

    	\(\sqrt{{x}^{3}}\)
    Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{{x}^{2}\cdot x}\)
    Rewrite the radical as the product of two radicals.	\(\sqrt{{x}^{2}}\cdot \sqrt{x}\)
    Simplify.	\(x\sqrt{x}\)

11. Simplify: \(\sqrt{{b}^{5}}\).

    通过它工作 提示

    发送答案

    \({b}^{2}\sqrt{b}\)

12. Simplify: \(\sqrt{{p}^{9}}\).

    通过它工作 提示

    发送答案

    \({p}^{4}\sqrt{p}\)

13. Simplify: \(\sqrt{25{y}^{5}.}\)

    通过它工作 提示

    发送答案

    	\(\sqrt{25{y}^{5}}\)
    Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{25{y}^{4}\cdot y}\)
    Rewrite the radical as the product of two radicals.	\(\sqrt{25{y}^{4}}\cdot \sqrt{y}\)
    Simplify.	\(5{y}^{2}\sqrt{y}\)

14. Simplify: \(\sqrt{16{x}^{7}}\).

    通过它工作 提示

    发送答案

    \(4{x}^{3}\sqrt{x}\)

15. Simplify: \(\sqrt{49{v}^{9}}\).

    通过它工作 提示

    发送答案

    \(7{v}^{4}\sqrt{v}\)

16. Simplify: \(\sqrt{72{n}^{7}}\).

    通过它工作 提示

    发送答案

    	\(\sqrt{72{n}^{7}}\)
    Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{36{n}^{6}\cdot 2n}\)
    Rewrite the radical as the product of two radicals.	\(\sqrt{36{n}^{6}}\cdot \sqrt{2n}\)
    Simplify.	\(6{n}^{3}\sqrt{2n}\)

17. Simplify: \(\sqrt{32{y}^{5}}\).

    通过它工作 提示

    发送答案

    \(4{y}^{2}\sqrt{2y}\)

18. Simplify: \(\sqrt{75{a}^{9}}\).

    通过它工作 提示

    发送答案

    \(5{a}^{4}\sqrt{3a}\)

19. Simplify: \(\sqrt{63{u}^{3}{v}^{5}}\).

    通过它工作 提示

    发送答案

    	\(\sqrt{63{u}^{3}{v}^{5}}\)
    Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{9{u}^{2}{v}^{4}\cdot 7uv}\)
    Rewrite the radical as the product of two radicals.	\(\sqrt{9{u}^{2}{v}^{4}}\cdot \sqrt{7uv}\)
    Simplify.	\(3u{v}^{2}\sqrt{7uv}\)

20. Simplify: \(\sqrt{98{a}^{7}{b}^{5}}\).

    通过它工作 提示

    发送答案

    \(7{a}^{3}{b}^{2}{\sqrt{2ab}}^{}\)

21. Simplify: \(\sqrt{180{m}^{9}{n}^{11}}\).

    通过它工作 提示

    发送答案

    \(6{m}^{4}{n}^{5}\sqrt{5mn}\)

22. Simplify: \(3+\sqrt{32}\).

    通过它工作 提示

    发送答案

    	\(3+\sqrt{32}\)
    Rewrite the radicand as a product using the largest perfect square factor.	\(3+\sqrt{16\cdot 2}\)
    Rewrite the radical as the product of two radicals.	\(3+\sqrt{16}\cdot \sqrt{2}\)
    Simplify.	\(3+4\sqrt{2}\)

    The terms are not like and so we cannot add them. Trying to add an integer and a radical is like trying to add an integer and a variable, they are not like terms!

23. Simplify: \(5+\sqrt{75}\).

    通过它工作 提示

    发送答案

    \(5+5\sqrt{3}\)

24. Simplify: \(2+\sqrt{98}\).

    通过它工作 提示

    发送答案

    \(2+7\sqrt{2}\)

25. Simplify: \(\frac{4-\sqrt{48}}{2}\).

    通过它工作 提示

    发送答案

    	\(\frac{4-\sqrt{48}}{2}\)
    Rewrite the radicand as a product using the largest perfect square factor.	\(\frac{4-\sqrt{16\cdot 3}}{2}\)
    Rewrite the radical as the product of two radicals.	\(\frac{4-\sqrt{16}\cdot \sqrt{3}}{2}\)
    Simplify.	\(\frac{4-4\sqrt{3}}{2}\)
    Factor the common factor from the numerator.	\(\frac{4(1-\sqrt{3})}{2}\)
    Remove the common factor, 2, from the numerator and denominator.	\(\frac{2\cdot 2(1-\sqrt{3})}{2}\)
    Simplify.	\(2(1-\sqrt{3})\)

26. Simplify: \(\frac{10-\sqrt{75}}{5}\).

    通过它工作 提示

    发送答案

    \(2-\sqrt{3}\)

27. Simplify: \(\frac{6-\sqrt{45}}{3}\).

    通过它工作 提示

    发送答案

    \(2-\sqrt{5}\)

28. Simplify: \(\sqrt{\frac{9}{64}}\).

    通过它工作 提示

    发送答案

    \(\begin{array}{llll} & & & \ \sqrt{\frac{9}{64}} \\ \text{Since}\ {(\frac{3}{8})}^{2}=\frac{9}{64} & & & \ \frac{3}{8}\end{array}\)

29. Simplify: \(\sqrt{\frac{25}{16}}\).

    通过它工作 提示

    发送答案

    \(\frac{5}{4}\)

30. Simplify: \(\sqrt{\frac{49}{81}}\).

    通过它工作 提示

    发送答案

    \(\frac{7}{9}\)

31. Simplify: \(\sqrt{\frac{45}{80}}\).

    通过它工作 提示

    发送答案

    	\(\sqrt{\frac{45}{80}}\)
    Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.	\(\sqrt{\frac{5\cdot 9}{5\cdot 16}}\)
    Simplify the fraction by removing common factors.	\(\sqrt{\frac{9}{16}}\)
    \(\text{Simplify.}\ {(\frac{3}{4})}^{2}=\frac{9}{16}\)	\(\frac{3}{4}\)

32. Simplify: \(\sqrt{\frac{75}{48}}\).

    通过它工作 提示

    发送答案

    \(\frac{5}{4}\)

33. Simplify: \(\sqrt{\frac{98}{162}}\).

    通过它工作 提示

    发送答案

    \(\frac{7}{9}\)

34. Simplify: \(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\).

    通过它工作 提示

    发送答案

    	\(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\)
    Simplify the fraction inside the radical first.
    Divide the like bases by subtracting the exponents.	\(\sqrt{{m}^{2}}\)
    Simplify.	\(m\)

35. Simplify: \(\sqrt{\frac{{a}^{8}}{{a}^{6}}}\).

    通过它工作 提示

    发送答案

    \(a\)

36. Simplify: \(\sqrt{\frac{{x}^{14}}{{x}^{10}}}\).

    通过它工作 提示

    发送答案

    \({x}^{2}\)

37. Simplify: \(\sqrt{\frac{48{p}^{7}}{3{p}^{3}}}\).

    通过它工作 提示

    发送答案

    	\(\sqrt{\frac{48{p}^{7}}{3{p}^{3}}}\)
    Simplify the fraction inside the radical first.	\(\sqrt{16{p}^{4}}\)
    Simplify.	\(4{p}^{2}\)

38. Simplify: \(\sqrt{\frac{75{x}^{5}}{3x}}\).

    通过它工作 提示

    发送答案

    \(5{x}^{2}\)

39. Simplify: \(\sqrt{\frac{72{z}^{12}}{2{z}^{10}}}\).

    通过它工作 提示

    发送答案

    \(6z\)

40. Simplify: \(\sqrt{\frac{21}{64}}\).

    通过它工作 提示

    发送答案

    	\(\sqrt{\frac{21}{64}}\)
    We cannot simplify the fraction inside the radical. Rewrite using the quotient property.	\(\frac{\sqrt{21}}{\sqrt{64}}\)
    Simplify the square root of 64. The numerator cannot be simplified.	\(\frac{\sqrt{21}}{8}\)

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\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.\neqnot equal
The two sides are different.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

如何: Simplify Square Roots

1. Use the Product Property to simplify square roots
2. Use the Quotient Property to simplify square roots
3. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
4. Use the product rule to rewrite the radical as the product of two radicals.
5. Simplify the square root of the perfect square.
6. Simplify the fraction in the radicand, if possible.
7. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
8. Simplify the radicals in the numerator and the denominator.

人们对人们提出的问题

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.

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