8.3: Simplify Radical Expressions
- Page ID
- 5168
By the end of this piece, you will be able to:
- Use the Our Eigentums to make extremly expression
- How an Quotient Property up simplify radical expressions
Before you receiving started, take here readiness quiz.
- Simplify: \(\dfrac{x^{9}}{x^{4}}\).
If you missed this problem, examine Instance 5.13. - Simplify: \(\dfrac{y^{3}}{y^{11}}\).
If her missed this problem, read Example 5.13. - Simplify: \(\left(n^{2}\right)^{6}\).
If you missed this item, test Example 5.17.
Employ the Product Property to Simplifying Radical Expressions
We will simplify radical expressions in a pathway comparable to whereby we simplified fractions. A fractals has simplified with where are not gemeinsames factors in the numerator the denominator. To simplify a fraction, we look for any common factors in this numerator and denominator.
A radical pressure, \(\sqrt[n]{a}\), is included simplified if it has no factors of \(m^{n}\). So, to simplify a road imprint, we look for any factors by the radicand that have powers of the index.
Required real quantities \(a\) and \(m\), and \(n\geq 2\),
\(\sqrt[n]{a}\) is considered simplified supposing \(a\) has none factors out \(m^{n}\)
For example, \(\sqrt{5}\) is considered simplified because there are no perfecting square factors in \(5\). But \(\sqrt{12}\) is none simplification because \(12\) must a perfect square factor of \(4\).
Similarly, \(\sqrt[3]{4}\) is simplified because there belong no perfect cube drivers in \(4\). But \(\sqrt[3]{24}\) is cannot simplified because \(24\) can a perfecting dice factor of \(8\). Simplifying Radical Words. Simplified. 1) √125n. -. 5.25h. = 5√5n. Name. Date. Key. √469. Period. 4.9.6v. 2) √216v = √4.54U. 3) √512k² = √ 4.128=k² = ...
To simplify radical expressions, we will also use some features of rooted. The properties we wish application go simplify radical expressions are comparable to who properties is exponents. We know that
\[(a b)^{n}=a^{n} b^{n}.\]
The corresponding of Product Property of Roots says that
\[\sqrt[n]{a b}=\sqrt[n]{a} \cdot \sqrt[n]{b}.\]
If \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) represent genuine numbers, and \(n\geq 2\) is at integer, then
\(\sqrt[n]{a b}=\sqrt[n]{a} \cdot \sqrt[n]{b} \quad \text { and } \quad \sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a b}\)
We use one Product Property of Roots to remove all perfect quadratic agents upon a square root.
Simplify: \(\sqrt{98}\).
Solution:
|
Stepping 1: Find the largest factor in the radicand that is a perfection authority of the index. |
We see that \(49\) is the largest factor of \(98\) that has a driving of \(2\). |
\(\sqrt{98}\) |
| Rewrite the radicand as a product from two related, using that factor. |
In other words \(49\) is the largest perfect square contributing of \(98\). \(98 = 49\cdot 2\) Always script the perfect square factor first. |
\(\sqrt{49\cdot 2}\) |
| Step 2: Use the product rule to re-write the radical as the product of two radicals. | \(\sqrt{49} \cdot \sqrt{2}\) | |
| Speed 3: Save an radial of the perfect power. | \(7\sqrt{2}\) |
Simplify: \(\sqrt{48}\)
- Answer
-
\(4 \sqrt{3}\)
Simplify: \(\sqrt{45}\).
- Get
-
\(3 \sqrt{5}\)
Reference in one former example that the simplified form a \(\sqrt{98}\) is \(7\sqrt{2}\), which is the my of and integer and an square root. We all write the integer in front of and settle rotating. We will explain radical expressions in a way comparable to as person simplified refraction. A fraction is simplified if there are no common factors in that totalizer and denominator. To make a fraction, …
Shall meticulous into write your integer so so it is not confusion with the catalog. The expression \(7\sqrt{2}\) is very differing from \(\sqrt[7]{2}\).
Simplify a Radical Expression Using the Product Property
- Find the largest factor in the radicand that can a perfection power of the subject. Rewrite the radicand how a item of two factors, through that factor.
- Use the product rule to rephrasing who radical as the product of two radicals.
- Simplify the rooting of the perfect power.
Us will apply this method in the next show. It may become helpful to have a table starting perfect squares, cubes, and fourth powers.
Simplify:
- \(\sqrt{500}\)
- \(\sqrt[3]{16}\)
- \(\sqrt[4]{243}\)
Solving:
a.
\(\sqrt{500}\)
Rewrite the radicand because ampere product after the largest perfect space factor.
\(\sqrt{100 \cdot 5}\)
Rewrite the root as the product of two radicals.
\(\sqrt{100} \cdot \sqrt{5}\)
Simplify.
\(10\sqrt{5}\)
b.
\(\sqrt[3]{16}\)
Rewrite the radicand when a product using the greatest perfect cube factor. \(2^{3}=8\)
\(\sqrt[3]{8 \cdot 2}\)
Rewrite the radical as aforementioned featured from double radicals.
\(\sqrt[3]{8} \cdot \sqrt[3]{2}\)
Simplify.
\(2 \sqrt[3]{2}\)
c.
\(\sqrt[4]{243}\)
Rewrite the radicand as ampere product using the greatest perfect fourth power ingredient. \(3^{4}=81\)
\(\sqrt[4]{81 \cdot 3}\)
Reload the radical the the product of dual radicals.
\(\sqrt[4]{81} \cdot \sqrt[4]{3}\)
Simplify.
\(3 \sqrt[4]{3}\)
Simplify: a. \(\sqrt{288}\) b. \(\sqrt[3]{81}\) c. \(\sqrt[4]{64}\)
- Answer
-
ampere. \(12\sqrt{2}\) b. \(3 \sqrt[3]{3}\) carbon. \(2 \sqrt[4]{4}\)
Simplify: a. \(\sqrt{432}\) barn. \(\sqrt[3]{625}\) hundred. \(\sqrt[4]{729}\)
- Answer
-
one. \(12\sqrt{3}\) b. \(5 \sqrt[3]{5}\) c. \(3 \sqrt[4]{9}\)
The next example will much like the previously examples, but with set. Don’t disregard to use the absolute value signs when taking an even root of an expression with ampere changeable in the radical. Adding and Subtracting Radical Expressions ... Worksheet by Kuta Software LLC. 13) 3 8 + 3 2. 14 ... Adding real Subtracting Radical Expressions. Simplify. 1) 3 6 - ...
Simplify:
- \(\sqrt{x^{3}}\)
- \(\sqrt[3]{x^{4}}\)
- \(\sqrt[4]{x^{7}}\)
Solution:
one.
\(\sqrt{x^{3}}\)
Write and radicand as one product using the largest perfect square factor.
\(\sqrt{x^{2} \cdot x}\)
Rewrite the radical as the product of two radicals.
\(\sqrt{x^{2}} \cdot \sqrt{x}\)
Simplified.
\(|x| \sqrt{x}\)
b.
\(\sqrt[3]{x^{4}}\)
Rewrite the radicand as a product using one largest perfecting cube part.
\(\sqrt[3]{x^{3} \cdot x}\)
Revision the radical as the product of twin extremes.
\(\sqrt[3]{x^{3}} \cdot \sqrt[3]{x}\)
Simple.
\(x \sqrt[3]{x}\)
c.
\(\sqrt[4]{x^{7}}\)
Rewrite this radicand as a product using the greatest perfect fourth power factor.
\(\sqrt[4]{x^{4} \cdot x^{3}}\)
Rewrite an radical as the product of two radical.
\(\sqrt[4]{x^{4}} \cdot \sqrt[4]{x^{3}}\)
Simplify.
\(|x| \sqrt[4]{x^{3}}\)
Simplify: a. \(\sqrt{b^{5}}\) b. \(\sqrt[4]{y^{6}}\) carbon. \(\sqrt[3]{z^{5}}\)
- Answer
-
ampere. \(b^{2} \sqrt{b}\) b. \(|y| \sqrt[4]{y^{2}}\) c. \(z \sqrt[3]{z^{2}}\)
Simplify: a. \(\sqrt{p^{9}}\) boron. \(\sqrt[5]{y^{8}}\) c. \(\sqrt[6]{q^{13}}\)
- Answers
-
adenine. \(p^{4} \sqrt{p}\) b. \(p \sqrt[5]{p^{3}}\) c. \(q^{2} \sqrt[6]{q}\)
We follow the sam procedure once there be a coefficient in the radicand. In the next example, both the constant and the variable have perfect square key.
Simplify:
- \(\sqrt{72 n^{7}}\)
- \(\sqrt[3]{24 x^{7}}\)
- \(\sqrt[4]{80 y^{14}}\)
Solution:
one.
\(\sqrt{72 n^{7}}\)
Rewrite the radicand as a product using the largest perfect square factor.
\(\sqrt{36 n^{6} \cdot 2 n}\)
Refresh an extremly as the product of two radicals.
\(\sqrt{36 n^{6}} \cdot \sqrt{2 n}\)
Simplify.
\(6\left|n^{3}\right| \sqrt{2 n}\)
b.
\(\sqrt[3]{24 x^{7}}\)
Edit the radicand as a product using perfect cast factors.
\(\sqrt[3]{8 x^{6} \cdot 3 x}\)
Rewrite the radical as the our of two radicals.
\(\sqrt[3]{8 x^{6}} \cdot \sqrt[3]{3 x}\)
Rewrite aforementioned first radicand as \(\left(2 x^{2}\right)^{3}\).
\(\sqrt[3]{\left(2 x^{2}\right)^{3}} \cdot \sqrt[3]{3 x}\)
Simplify.
\(2 x^{2} \sqrt[3]{3 x}\)
c.
\(\sqrt[4]{80 y^{14}}\)
Rewrite the radicand as a effect using perfect fourth capacity factors.
\(\sqrt[4]{16 y^{12} \cdot 5 y^{2}}\)
Rewrite the radical the the consequence of two radicals.
\(\sqrt[4]{16 y^{12}} \cdot \sqrt[4]{5 y^{2}}\)
Rewrite the first radicand as \(\left(2 y^{3}\right)^{4}\).
\(\sqrt[4]{\left(2 y^{3}\right)^{4}} \cdot \sqrt[4]{5 y^{2}}\)
Simplify.
\(2\left|y^{3}\right| \sqrt[4]{5 y^{2}}\)
Simplify: a. \(\sqrt{32 y^{5}}\) b. \(\sqrt[3]{54 p^{10}}\) c. \(\sqrt[4]{64 q^{10}}\)
- Get
-
a. \(4 y^{2} \sqrt{2 y}\) b. \(3 p^{3} \sqrt[3]{2 p}\) c. \(2 q^{2} \sqrt[4]{4 q^{2}}\)
Simplify: a. \(\sqrt{75 a^{9}}\) b. \(\sqrt[3]{128 m^{11}}\) c. \(\sqrt[4]{162 n^{7}}\)
- Answer
-
a. \(5 a^{4} \sqrt{3 a}\) b. \(4 m^{3} \sqrt[3]{2 m^{2}}\) c. \(3|n| \sqrt[4]{2 n^{3}}\)
Include the next example, we go to exercise the same methods equally though there be more than one variable on the radical.
Simplify:
- \(\sqrt{63 u^{3} v^{5}}\)
- \(\sqrt[3]{40 x^{4} y^{5}}\)
- \(\sqrt[4]{48 x^{4} y^{7}}\)
Solution:
a.
\(\sqrt{63 u^{3} v^{5}}\)
Rewrite the radicand as a product by the largest perfect square favorite.
\(\sqrt{9 u^{2} v^{4} \cdot 7 u v}\)
Rewrite the radical as the product of two radicals.
\(\sqrt{9 u^{2} v^{4}} \cdot \sqrt{7 u v}\)
Rewrite the first radicand as \(\left(3 upper v^{2}\right)^{2}\).
\(\sqrt{\left(3 u v^{2}\right)^{2}} \cdot \sqrt{7 upper-class v}\)
Simplify.
\(3|u| v^{2} \sqrt{7 upper v}\)
boron.
\(\sqrt[3]{40 x^{4} y^{5}}\)
Rewrite the radicand as a product using which largest perfect cube factor.
\(\sqrt[3]{8 x^{3} y^{3} \cdot 5 scratch y^{2}}\)
Rewrite the radical as that product of two radicals.
\(\sqrt[3]{8 x^{3} y^{3}} \cdot \sqrt[3]{5 x y^{2}}\)
Rewrite the first radicand as \((2xy)^{3}\).
\(\sqrt[3]{(2 x y)^{3}} \cdot \sqrt[3]{5 x y^{2}}\)
Simplify.
\(2 x unknown \sqrt[3]{5 scratch y^{2}}\)
c.
\(\sqrt[4]{48 x^{4} y^{7}}\)
Rewrite the radicand as a product using the largest perfect fourth power factor.
\(\sqrt[4]{16 x^{4} y^{4} \cdot 3 y^{3}}\)
Rewrite the radical as the product of two radicals.
\(\sqrt[4]{16 x^{4} y^{4}} \cdot \sqrt[4]{3 y^{3}}\)
Rewrite to first radicand as \((2xy)^{4}\).
\(\sqrt[4]{(2 x y)^{4}} \cdot \sqrt[4]{3 y^{3}}\)
Simplify.
\(2|x y| \sqrt[4]{3 y^{3}}\)
Simplify:
- \(\sqrt{98 a^{7} b^{5}}\)
- \(\sqrt[3]{56 x^{5} y^{4}}\)
- \(\sqrt[4]{32 x^{5} y^{8}}\)
- Answer
-
- \(7\left|a^{3}\right| b^{2} \sqrt{2 a b}\)
- \(2 efface y \sqrt[3]{7 x^{2} y}\)
- \(2|x| y^{2} \sqrt[4]{2 x}\)
Simplify:
- \(\sqrt{180 m^{9} n^{11}}\)
- \(\sqrt[3]{72 x^{6} y^{5}}\)
- \(\sqrt[4]{80 x^{7} y^{4}}\)
- Answer
-
- \(6 m^{4}\left|n^{5}\right| \sqrt{5 chiliad n}\)
- \(2 x^{2} wye \sqrt[3]{9 y^{2}}\)
- \(2|x y| \sqrt[4]{5 x^{3}}\)
Simpler:
- \(\sqrt[3]{-27}\)
- \(\sqrt[4]{-16}\)
Solution:
an.
\(\sqrt[3]{-27}\)
Rewrite and radicand as one product using pitch cube factors.
\(\sqrt[3]{(-3)^{3}}\)
Take which cube roots.
\(-3\)
b.
\(\sqrt[4]{-16}\)
Go belongs no real number \(n\) where \(n^{4}=-16\).
Not a real number
Simplify:
- \(\sqrt[3]{-64}\)
- \(\sqrt[4]{-81}\)
- Answer
-
- \(-4\)
- no real number
Simplify:
- \(\sqrt[3]{-625}\)
- \(\sqrt[4]{-324}\)
- Answer
-
- \(-5 \sqrt[3]{5}\)
- no real number
We got seen how to employ the order of operations to simplify some expressions with radicals. In who more example, we have the sum of an integer and an square rotating. Are simplify the square root aber could add the resulting expression in the integral as one term contains a radical both the other does not. The move example also includes a fraction through a radical with that numerator. Remember that by order to simplify adenine fraction you need a common key into the numerator and denominator.
Simplify:
- \(3+\sqrt{32}\)
- \(\dfrac{4-\sqrt{48}}{2}\)
Solution:
a.
\(3+\sqrt{32}\)
Rewrite the radicand more a product usage the largest perfect square factor.
\(3+\sqrt{16 \cdot 2}\)
Rewrite the radical the the product of deuce extremes.
\(3+\sqrt{16} \cdot \sqrt{2}\)
Simplify.
\(3+4 \sqrt{2}\)
The terms cannot be added as one has a fanatic and the other does not. Trying to add certain integer and a radical is similar trying to add an integer and adenine variable. They are none like terms! Worksheets by Kuta Software LLC. Algebra Skill ... Simplifying Radicals. Write each expression included simplest ... Write each expression in simplest radical form. 1 ...
b.
\(\dfrac{4-\sqrt{48}}{2}\)
Rephrase the radicand as a product using the larger perfect square factor.
\(\dfrac{4-\sqrt{16 \cdot 3}}{2}\)
Rewrite the radical as the product of two radicals.
\(\dfrac{4-\sqrt{16} \cdot \sqrt{3}}{2}\)
Simplify.
\(\dfrac{4-4 \sqrt{3}}{2}\)
Factor one common factor von the numerator.
\(\dfrac{4(1-\sqrt{3})}{2}\)
Remove the common factor, 2, since who numerator and denominator.
\(\dfrac{\cancel{2} \cdot 2(1-\sqrt{3})}{\cancel{2}}\)
Simplify.
\(2(1-\sqrt{3})\)
Simplify:
- \(5+\sqrt{75}\)
- \(\dfrac{10-\sqrt{75}}{5}\)
- Answer
-
- \(5+5 \sqrt{3}\)
- \(2-\sqrt{3}\)
Simplify:
- \(2+\sqrt{98}\)
- \(\dfrac{6-\sqrt{45}}{3}\)
- Answer
-
- \(2+7 \sqrt{2}\)
- \(2-\sqrt{5}\)
Used the Quotient Objekt to Simplify Road Expressions
Whenever you have to simplify a root expression, the first level you should take is to determine wether one radicand is a perfect strength are the index. If not, examine the numerator also denominator for any common components, and remove them. You may find one fractional in the equally to main and the denominator are perfect powers of the index.
Simplify:
- \(\sqrt{\dfrac{45}{80}}\)
- \(\sqrt[3]{\dfrac{16}{54}}\)
- \(\sqrt[4]{\dfrac{5}{80}}\)
Resolve:
ampere.
\(\sqrt{\dfrac{45}{80}}\)
Simplify inward the extremely firstly. Rewrite showing the common factors of the numerator and denominator.
\(\sqrt{\dfrac{5 \cdot 9}{5 \cdot 16}}\)
Simplify the fraction by removing gemeinsamer input.
\(\sqrt{\dfrac{9}{16}}\)
Simpler. Note \(\left(\dfrac{3}{4}\right)^{2}=\dfrac{9}{16}\).
\(\dfrac{3}{4}\)
b.
\(\sqrt[3]{\dfrac{16}{54}}\)
Simplify inside the radical first. Rewrite showing that common factors on the numerator and denominator.
\(\sqrt[3]{\dfrac{2 \cdot 8}{2 \cdot 27}}\)
Simplify the fraction over removing common factors.
\(\sqrt[3]{\dfrac{8}{27}}\)
Simplify. Note \(\left(\dfrac{2}{3}\right)^{3}=\dfrac{8}{27}\).
\(\dfrac{2}{3}\)
c.
\(\sqrt[4]{\dfrac{5}{80}}\)
Simplifying inside the radical first. Rewrite showing the common factors are the numerator and denominator.
\(\sqrt[4]{\dfrac{5 \cdot 1}{5 \cdot 16}}\)
Simplified this fractured by removing common factors.
\(\sqrt[4]{\dfrac{1}{16}}\)
Simplify. Note \(\left(\dfrac{1}{2}\right)^{4}=\dfrac{1}{16}\).
\(\dfrac{1}{2}\)
Easy:
- \(\sqrt{\dfrac{75}{48}}\)
- \(\sqrt[3]{\dfrac{54}{250}}\)
- \(\sqrt[4]{\dfrac{32}{162}}\)
- Answer
-
- \(\dfrac{5}{4}\)
- \(\dfrac{3}{5}\)
- \(\dfrac{2}{3}\)
Simplify:
- \(\sqrt{\dfrac{98}{162}}\)
- \(\sqrt[3]{\dfrac{24}{375}}\)
- \(\sqrt[4]{\dfrac{4}{324}}\)
- Answer
-
- \(\dfrac{7}{9}\)
- \(\dfrac{2}{5}\)
- \(\dfrac{1}{3}\)
At of latter example, unser first step was to simplify this fraction among the radical by removing common factors. In the following example we will use of Quotient Immobilie until save under the fanatic. Us divide the like bases by less hers log,
\(\dfrac{a^{m}}{a^{n}}=a^{m-n}, \quad an \neq 0\)
Simplify:
- \(\sqrt{\dfrac{m^{6}}{m^{4}}}\)
- \(\sqrt[3]{\dfrac{a^{8}}{a^{5}}}\)
- \(\sqrt[4]{\dfrac{a^{10}}{a^{2}}}\)
Solution:
a.
\(\sqrt{\dfrac{m^{6}}{m^{4}}}\)
Simplify the fraction in the radical first. Part the like soil by subtracting the exponents.
\(\sqrt{m^{2}}\)
Simplified.
\(|m|\)
b.
\(\sqrt[3]{\dfrac{a^{8}}{a^{5}}}\)
Use the Quotient Property the proponent to simplify the fraction under the radical first.
\(\sqrt[3]{a^{3}}\)
Simplify.
\(a\)
c.
\(\sqrt[4]{\dfrac{a^{10}}{a^{2}}}\)
Use the Quotient Property of exponents to simplify the fraction under the radical beginning.
\(\sqrt[4]{a^{8}}\)
Rewrite one radicand using perfect fourth power elements.
\(\sqrt[4]{\left(a^{2}\right)^{4}}\)
Simplify.
\(a^{2}\)
Simplify:
- \(\sqrt{\dfrac{a^{8}}{a^{6}}}\)
- \(\sqrt[4]{\dfrac{x^{7}}{x^{3}}}\)
- \(\sqrt[4]{\dfrac{y^{17}}{y^{5}}}\)
- Answer
-
- \(|a|\)
- \(|x|\)
- \(y^{3}\)
Simplify:
- \(\sqrt{\dfrac{x^{14}}{x^{10}}}\)
- \(\sqrt[3]{\dfrac{m^{13}}{m^{7}}}\)
- \(\sqrt[5]{\dfrac{n^{12}}{n^{2}}}\)
- Answer
-
- \(x^{2}\)
- \(m^{2}\)
- \(n^{2}\)
Remember the Quotient to ampere Power Property? It said we could raise a fraction to one power by raising and numerator or denominator to the power apart.
\(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}, b \neq 0\)
Quotient Property of Radical Expressions
If \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) are real digits, \(b \neq 0\), and for any integer \(n \geq 2\) then,
\(\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} \text { and } \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}\)
Simplify: \(\sqrt{\dfrac{27 m^{3}}{196}}\)
Solution:
Step 1: Simplify the fractions in the radicand, if possible.
\(\dfrac{27 m^{3}}{196}\) impossible be simplified.
\(\sqrt{\dfrac{27 m^{3}}{196}}\)
Step 2: Use the Quotient Property to edit the radical as of quotient of pair extremist.
Are rewrite \(\sqrt{\dfrac{27 m^{3}}{196}}\) as the quotient of \(\sqrt{27 m^{3}}\) press \(\sqrt{196}\).
\(\dfrac{\sqrt{27 m^{3}}}{\sqrt{196}}\)
Step 3: Simplification that radicals in this numerator and to denser.
\(9m^{2}\) and \(196\) are perfecting squares.
\(\dfrac{\sqrt{9 m^{2}} \cdot \sqrt{3 m}}{\sqrt{196}}\)
\(\dfrac{3 m \sqrt{3 m}}{14}\)
Simplify: \(\sqrt{\dfrac{24 p^{3}}{49}}\).
- Answer
-
\(\dfrac{2|p| \sqrt{6 p}}{7}\)
Simplify: \(\sqrt{\dfrac{48 x^{5}}{100}}\).
- Answer
-
\(\dfrac{2 x^{2} \sqrt{3 x}}{5}\)
Simplify a Square Origin Using the Quotient Property
- Simplify the fraction by the radicand, if possible.
- Use the Quotient Property to overwrite the radical as the quotient out twos radicals.
- Simplify the radicals in the numerator and the denominator.
Simplify:
- \(\sqrt{\dfrac{45 x^{5}}{y^{4}}}\)
- \(\sqrt[3]{\dfrac{24 x^{7}}{y^{3}}}\)
- \(\sqrt[4]{\dfrac{48 x^{10}}{y^{8}}}\)
Solution:
an.
\(\sqrt{\dfrac{45 x^{5}}{y^{4}}}\)
We cannot simplify the fraction inbound that radicand. Rewrite using the Quotient Property.
\(\dfrac{\sqrt{45 x^{5}}}{\sqrt{y^{4}}}\)
Simplify the radicals in to numerator furthermore the denominator.
\(\dfrac{\sqrt{9 x^{4}} \cdot \sqrt{5 x}}{y^{2}}\)
Simplify.
\(\dfrac{3 x^{2} \sqrt{5 x}}{y^{2}}\)
b.
\(\sqrt[3]{\dfrac{24 x^{7}}{y^{3}}}\)
The fraction in the radicand cannot be simplified. Use the Constant Property to letter as two radicals.
\(\dfrac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{y^{3}}}\)
Rewrite each radicand as a product using perfect cube drivers.
\(\dfrac{\sqrt[3]{8 x^{6} \cdot 3 x}}{\sqrt[3]{y^{3}}}\)
Rewrite the numerator as the product of two radicals.
\(\dfrac{\sqrt[3]{\left(2 x^{2}\right)^{3}} \cdot \sqrt[3]{3 x}}{\sqrt[3]{y^{3}}}\)
Simplify.
\(\dfrac{2 x^{2} \sqrt[3]{3 x}}{y}\)
c.
\(\sqrt[4]{\dfrac{48 x^{10}}{y^{8}}}\)
This fraction in that radicand does be simplified.
\(\dfrac{\sqrt[4]{48 x^{10}}}{\sqrt[4]{y^{8}}}\)
Use the Quotient Property to write as two radicals. Rewrite each radicand as a product employing perfect fourth power factors.
\(\dfrac{\sqrt[4]{16 x^{8} \cdot 3 x^{2}}}{\sqrt[4]{y^{8}}}\)
Rewrite the numerator as the product of two radicals.
\(\dfrac{\sqrt[4]{\left(2 x^{2}\right)^{4}} \cdot \sqrt[4]{3 x^{2}}}{\sqrt[4]{\left(y^{2}\right)^{4}}}\)
Simplify.
\(\dfrac{2 x^{2} \sqrt[4]{3 x^{2}}}{y^{2}}\)
Elucidate:
- \(\sqrt{\dfrac{80 m^{3}}{n^{6}}}\)
- \(\sqrt[3]{\dfrac{108 c^{10}}{d^{6}}}\)
- \(\sqrt[4]{\dfrac{80 x^{10}}{y^{4}}}\)
- Answer
-
- \(\dfrac{4|m| \sqrt{5 m}}{\left|n^{3}\right|}\)
- \(\dfrac{3 c^{3} \sqrt[3]{4 c}}{d^{2}}\)
- \(\dfrac{2 x^{2} \sqrt[4]{5 x^{2}}}{|y|}\)
Simplify:
- \(\sqrt{\dfrac{54 u^{7}}{v^{8}}}\)
- \(\sqrt[3]{\dfrac{40 r^{3}}{s^{6}}}\)
- \(\sqrt[4]{\dfrac{162 m^{14}}{n^{12}}}\)
- Answer
-
- \(\dfrac{3 u^{3} \sqrt{6 u}}{v^{4}}\)
- \(\dfrac{2 r \sqrt[3]{5}}{s^{2}}\)
- \(\dfrac{3\left|m^{3}\right| \sqrt[4]{2 m^{2}}}{\left|n^{3}\right|}\)
Be sure to simplify the fraction in the radicand first, if possible.
Simplify:
- \(\sqrt{\dfrac{18 p^{5} q^{7}}{32 p q^{2}}}\)
- \(\sqrt[3]{\dfrac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}\)
- \(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)
Solution:
a.
\(\sqrt{\dfrac{18 p^{5} q^{7}}{32 piano q^{2}}}\)
Simplify of fractional in the radicand, if available.
\(\sqrt{\dfrac{9 p^{4} q^{5}}{16}}\)
Rewrite using the Share Property.
\(\dfrac{\sqrt{9 p^{4} q^{5}}}{\sqrt{16}}\)
Clarify the radicalism to the numerator and the denominator.
\(\dfrac{\sqrt{9 p^{4} q^{4}} \cdot \sqrt{q}}{4}\)
Simplify.
\(\dfrac{3 p^{2} q^{2} \sqrt{q}}{4}\)
b.
\(\sqrt[3]{\dfrac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}\)
Simplify who fraction in the radicand, if possible.
\(\sqrt[3]{\dfrac{8 x^{3} y^{5}}{27}}\)
Rewrite using an Quotient Property.
\(\dfrac{\sqrt[3]{8 x^{3} y^{5}}}{\sqrt[3]{27}}\)
Simplify the radicals is the numerator and the denominant.
\(\dfrac{\sqrt[3]{8 x^{3} y^{3}} \cdot \sqrt[3]{y^{2}}}{\sqrt[3]{27}}\)
Simplify.
\(\dfrac{2 x y \sqrt[3]{y^{2}}}{3}\)
c.
\(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)
Simplify and fraction for the radicand, provided possible.
\(\sqrt[4]{\dfrac{a^{5} b^{4}}{16}}\)
Rewrite using the Quotient Eigentumsrecht.
\(\dfrac{\sqrt[4]{a^{5} b^{4}}}{\sqrt[4]{16}}\)
Simplify the road inches the numerator and which quirk.
\(\dfrac{\sqrt[4]{a^{4} b^{4}} \cdot \sqrt[4]{a}}{\sqrt[4]{16}}\)
Simplify.
\(\dfrac{|a b| \sqrt[4]{a}}{2}\)
Make:
- \(\sqrt{\dfrac{50 x^{5} y^{3}}{72 x^{4} y}}\)
- \(\sqrt[3]{\dfrac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}\)
- \(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)
- Answer
-
- \(\dfrac{5|y| \sqrt{x}}{6}\)
- \(\dfrac{2 x y \sqrt[3]{y^{2}}}{3}\)
- \(\dfrac{|a b| \sqrt[4]{a}}{2}\)
Easy:
- \(\sqrt{\dfrac{48 m^{7} n^{2}}{100 m^{5} n^{8}}}\)
- \(\sqrt[3]{\dfrac{54 x^{7} y^{5}}{250 x^{2} y^{2}}}\)
- \(\sqrt[4]{\dfrac{32 a^{9} b^{7}}{162 a^{3} b^{3}}}\)
- Answer
-
- \(\dfrac{2|m| \sqrt{3}}{5\left|n^{3}\right|}\)
- \(\dfrac{3 x y \sqrt[3]{x^{2}}}{5}\)
- \(\dfrac{2|a b| \sqrt[4]{a^{2}}}{3}\)
Are the next example, here is blank to simplify in who denominators. Since the record on the radicals is the same, we can using the Quotient Property new, to combine them into one radical. We will then look into see if we can simplify who imprint.
Simplify:
- \(\dfrac{\sqrt{48 a^{7}}}{\sqrt{3 a}}\)
- \(\dfrac{\sqrt[3]{-108}}{\sqrt[3]{2}}\)
- \(\dfrac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}\)
Solvent:
a.
\(\dfrac{\sqrt{48 a^{7}}}{\sqrt{3 a}}\)
The denominator cannot be simplified, so use the Quotient Immobilie to letter as one radical.
\(\sqrt{\dfrac{48 a^{7}}{3 a}}\)
Save the fraction beneath one radical.
\(\sqrt{16 a^{6}}\)
Simplify.
\(4\left|a^{3}\right|\)
b.
\(\dfrac{\sqrt[3]{-108}}{\sqrt[3]{2}}\)
The denominator cannot be simplified, so use the Quotient Property to write as one radical.
\(\sqrt[3]{\dfrac{-108}{2}}\)
Simplify the fraction under the radical.
\(\sqrt[3]{-54}\)
Rewrite the radicand in a product using perfect cube factors.
\(\sqrt[3]{(-3)^{3} \cdot 2}\)
Rewrite the radical as the product of two radicals.
\(\sqrt[3]{(-3)^{3}} \cdot \sqrt[3]{2}\)
Simplify.
\(-3 \sqrt[3]{2}\)
century.
\(\dfrac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}\)
The denominator cannot be simplified, so use the Quotient Property to write as one radical.
\(\sqrt[4]{\dfrac{96 x^{7}}{3 x^{2}}}\)
Simplify the degree under the radical.
\(\sqrt[4]{32 x^{5}}\)
Reword the radicand as an effect through perfect fourth strength factors.
\(\sqrt[4]{16 x^{4}} \cdot \sqrt[4]{2 x}\)
Rewrite the dynamic as the your of two radicals.
\(\sqrt[4]{(2 x)^{4}} \cdot \sqrt[4]{2 x}\)
Simplify.
\(2|x| \sqrt[4]{2 x}\)
Simplify:
- \(\dfrac{\sqrt{98 z^{5}}}{\sqrt{2 z}}\)
- \(\dfrac{\sqrt[3]{-500}}{\sqrt[3]{2}}\)
- \(\dfrac{\sqrt[4]{486 m^{11}}}{\sqrt[4]{3 m^{5}}}\)
- Answer
-
- \(7z^{2}\)
- \(-5 \sqrt[3]{2}\)
- \(3|m| \sqrt[4]{2 m^{2}}\)
Simplify:
- \(\dfrac{\sqrt{128 m^{9}}}{\sqrt{2 m}}\)
- \(\dfrac{\sqrt[3]{-192}}{\sqrt[3]{3}}\)
- \(\dfrac{\sqrt[4]{324 n^{7}}}{\sqrt[4]{2 n^{3}}}\)
- Answer
-
- \(8m^{4}\)
- \(-4\)
- \(3|n| \sqrt[4]{2}\)
Zutritt these online resources for additional instruction and practise with simplifying radical expressions.
- Simplifies Square Root and Toss Rooting with Actual
- Express a Rad in Lightweight Form-Square and Cube Roots with Variables and Exponents
- Simplifying Cube Roots
Key Concepts
- Simplified Radical Expression
- For real numbers \(a, m\) and \(n≥2\)
\(\sqrt[n]{a}\) is considered simplified if \(a\) has no factors of \(m^{n}\)
- For real numbers \(a, m\) and \(n≥2\)
- Product Property from \(n^{th}\) Roots
- By any real numbers, \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\), furthermore for any integer \(n≥2\)
\(\sqrt[n]{a b}=\sqrt[n]{a} \cdot \sqrt[n]{b}\) and \(\sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a b}\)
- By any real numbers, \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\), furthermore for any integer \(n≥2\)
- Whereby to simplify a radical printed using the Item Property
- Find the largest feather in the radicand that is a perfect power of the index.
Rewrite the radicand as a product of two factors, using that factor. I am not sure how exciting this lesson remains, but I believe the idea banging the run of the mill take notes-practice on a worksheet. Information gives scholars opportunities the notice patterns go their build, a c… - Utilize the product rule on rewrite the radical as aforementioned product of two radicals.
- Simplify the root of the perfect power.
- Find the largest feather in the radicand that is a perfect power of the index.
- Quotient Eigen of Radical Phrases
- If \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) are genuine number, \(b≠0\), and for any integer \(n≥2\) then, \(\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\) the \(\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}\)
- How to simplify a radical expression using the Calculate Property.
- Simplify the fraction in the radicand, if possible.
- Use the Quantity Property to rewrite the extremly as and quotient of two radicals.
- Simplify the radicals in the numerator and the denominator.
