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Simplify Expressions with Roots
Simplify expressions with roots
Simplify Expressions with Roots
In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write \({n}^{2}\) and read it ‘n squared’. This number is called the square of n, and n is called the square root. For example,
\[\begin{array}{l}{13}^{2}\ \text{is read “13 squared”} \\ \text{169 is called the}\ \text{square}\ \text{of 13, since}\ {13}^{2}=169 \\ \text{13 is a}\ \text{square root}\ \text{of 169}\end{array}\]Notice (−13)2 = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169.
So, every positive number has two square roots, one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write, \(\sqrt{m},\) which denotes the positive square root of m. The positive square root is also called the principal square root. This symbol, as well as other radicals to be introduced later, are grouping symbols.
We also use the radical sign for the square root of zero. Because \({0}^{2}=0,\) \(\sqrt{0}=0.\) Notice that zero has only one square root.
We know that every positive number has two square roots and the radical sign indicates the positive one. We write \(\sqrt{169}=13.\) If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, \(\text{-}\sqrt{169}=-13.\)
Example
Try it.
Simplify: ⓐ \(\sqrt{144}\) ⓑ \(\text{-}\sqrt{289}.\)
Solution
ⓐ
| \(\sqrt{144}\) | |
| Since \({12}^{2}=144.\) | \(12\) |
ⓑ
| \(-\sqrt{289}\) | |
| Since \({17}^{2}=289\) and the negative is in front of the radical sign. | \(-17\) |
Can we simplify \(\sqrt{-49}?\) Is there a number whose square is \(-49?\)
\[{(\ )}^{2}=-49\]Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to \(\sqrt{-49}.\) The square root of a negative number is not a real number.
Example
Try it.
Simplify: ⓐ \(\sqrt{-196}\) ⓑ \(\text{-}\sqrt{64}.\)
Solution
ⓐ
| \(\sqrt{-196}\) | |
| There is no real number whose square is \(-196.\) | \(\sqrt{-196}\ \text{is not a real number.}\) |
ⓑ
| \(-\sqrt{64}\) | |
| The negative is in front of the radical. | \(-8\) |
Condensed: the full section is in OpenStax Intermediate Algebra 2e.
Estimate and Approximate Roots
When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the \(\sqrt{4}=2,\) what is the value of \(\sqrt[]{21}\) or \(\sqrt[3]{50}?\) In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.
To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of \(\sqrt{11},\) we see 11 is between perfect square numbers 9 and 16, closer to 9. Its square root then will be between 3 and 4, but closer to 3.
Similarly, to estimate \(\sqrt[3]{91},\) we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.
Example
Try it.
Estimate each root between two consecutive whole numbers: ⓐ \(\sqrt{105}\) ⓑ \(\sqrt[3]{43}.\)
Solution
ⓐ Think of the perfect square numbers closest to 105. Make a small table of these perfect squares and their squares roots.
| Locate 105 between two consecutive perfect squares. | |
| \(\sqrt{105}\) is between their square roots. |
ⓑ Similarly we locate 43 between two perfect cube numbers.
| Locate 43 between two consecutive perfect cubes. | |
| \(\sqrt[3]{43}\) is between their cube roots. |
There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the \(\sqrt{x}\) key on your calculator. To find a cube root, or any root with higher index, you will use the \(\sqrt[y]{x}\) key.
When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is \(\approx\) and it is read ‘approximately’.
Suppose your calculator has a 10 digit display. You would see that
\[\begin{array}{l}\sqrt{5}\approx 2.236067978\ \text{rounded to two decimal places is}\ \sqrt{5}\approx 2.24 \\ \sqrt[4]{93}\approx 3.105422799\ \text{rounded to two decimal places is}\ \sqrt[4]{93}\approx 3.11\end{array}\]How do we know these values are approximations and not the exact values? Look at what happens when we square them:
\[\begin{array}{lll}{(2.236067978)}^{2} & = & 5.000000002 \\ {(2.24)}^{2} & = & 5.0176\end{array}\ \begin{array}{lll}{(3.105422799)}^{4} & = & 92.999999991 \\ {(3.11)}^{4} & = & 93.54951841\end{array}\]Condensed: the full section is in OpenStax Intermediate Algebra 2e.
Simplify Variable Expressions with Roots
The odd root of a number can be either positive or negative. For example,
But what about an even root? We want the principal root, so \(\sqrt[4]{625}=5.\)
But notice,
How can we make sure the fourth root of −5 raised to the fourth power is 5? We can use the absolute value. \(|-5|=5.\) So we say that when n is even \(\sqrt[n]{{a}^{n}}=|a|.\) This guarantees the principal root is positive.
Example
Try it.
Simplify: ⓐ \(\sqrt{{x}^{2}}\) ⓑ \(\sqrt[3]{{n}^{3}}\) ⓒ \(\sqrt[4]{{p}^{4}}\) ⓓ \(\sqrt[5]{{y}^{5}}.\)
Solution
ⓐ We use the absolute value to be sure to get the positive root.
| \(\sqrt{{x}^{2}}\) | |
| Since the index \(n\) is even, \(\sqrt[n]{{a}^{n}}=|a|.\) | \(|x|\) |
ⓑ This is an odd indexed root so there is no need for an absolute value sign.
| \(\sqrt[3]{{n}^{3}}\) | |
| Since the index \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\) | \(n\) |
ⓒ
| \(\sqrt[4]{{p}^{4}}\) | |
| Since the index \(n\ \text{is even}\ \sqrt[n]{{a}^{n}}=|a|.\) | \(|p|\) |
ⓓ
| \(\sqrt[5]{{y}^{5}}\) | |
| Since the index \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\) | \(y\) |
What about square roots of higher powers of variables? The Power Property of Exponents says \({({a}^{m})}^{n}={a}^{m\cdot n}.\) So if we square am, the exponent will become 2m.
\[{({a}^{m})}^{2}={a}^{2m}\]Looking now at the square root,
\(\begin{array}{llll} & & & \ \sqrt{{a}^{2m}} \\ \text{Since}\ {({a}^{m})}^{2}={a}^{2m}. & & & \ \sqrt{{({a}^{m})}^{2}} \\ \text{Since}\ n\ \text{is even}\ \sqrt[n]{{a}^{n}}=|a|. & & & \ |{a}^{m}| \\ & & & \ \text{So}\ \sqrt{{a}^{2m}}=|{a}^{m}|.\end{array}\)
Example
Try it.
Simplify: ⓐ \(\sqrt{{x}^{6}}\) ⓑ \(\sqrt{{y}^{16}}.\)
Solution
ⓐ
| \(\sqrt{{x}^{6}}\) | |
| Since \({({x}^{3})}^{2}={x}^{6}.\) | \(\sqrt{{({x}^{3})}^{2}}\) |
| Since the index \(n\) is even \(\sqrt{{a}^{n}}=|a|.\) | \(|{x}^{3}|\) |
ⓑ
| \(\sqrt{{y}^{16}}\) | |
| Since \({({y}^{8})}^{2}={y}^{16}.\) | \(\sqrt{{({y}^{8})}^{2}}\) |
| Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=|a|.\) | \({y}^{8}\) |
| In this case the absolute value sign is not needed as \({y}^{8}\) is positive. |
Condensed: the full section is in OpenStax Intermediate Algebra 2e.
Key Concepts
- Square Root Notation
- \(\sqrt{m}\) is read ‘the square root of m’
- If n2 = m, then \(n=\sqrt{m},\) for \(n\ge 0.\)
- The square root of m, \(\sqrt{m},\) is a positive number whose square is m.
- nth Root of a Number
- If \({b}^{n}=a,\) then b is an nth root of a.
- The principal nth root of a is written \(\sqrt[n]{a}.\)
- n is called the index of the radical.
- Properties of \(\sqrt[n]{a}\)
- When n is an even number and
- \(a\ge 0,\) then \(\sqrt[n]{a}\) is a real number
- \(a<0,\) then \(\sqrt[n]{a}\) is not a real number
- When n is an odd number, \(\sqrt[n]{a}\) is a real number for all values of a.
- When n is an even number and
- Simplifying Odd and Even Roots
- For any integer \(n\ge 2,\)
- when n is odd \(\sqrt[n]{{a}^{n}}=a\)
- when n is even \(\sqrt[n]{{a}^{n}}=|a|\)
- We must use the absolute value signs when we take an even root of an expression with a variable in the radical.
- For any integer \(n\ge 2,\)
Simplify Expressions with Roots
Simplify Expressions with Roots
In the following exercises, simplify.
Try it.
ⓐ \(\sqrt{64}\) ⓑ \(\text{-}\sqrt{81}\)
Solution
ⓐ 8 ⓑ \(-9\)
Try it.
ⓐ \(\sqrt{169}\) ⓑ \(\text{-}\sqrt{100}\)
Try it.
ⓐ \(\sqrt{196}\) ⓑ \(\text{-}\sqrt{1}\)
Solution
ⓐ 14 ⓑ \(-1\)
Try it.
ⓐ \(\sqrt{144}\) ⓑ \(\text{-}\sqrt{121}\)
Try it.
ⓐ \(\sqrt{\frac{4}{9}}\) ⓑ \(\text{-}\sqrt{0.01}\)
Solution
ⓐ \(\frac{2}{3}\) ⓑ \(-0.1\)
Try it.
ⓐ \(\sqrt{\frac{64}{121}}\) ⓑ \(\text{-}\sqrt{0.16}\)
Try it.
ⓐ \(\sqrt{-121}\) ⓑ \(\text{-}\sqrt{289}\)
Solution
ⓐ not real number ⓑ \(-17\)
Try it.
ⓐ \(\text{-}\sqrt{400}\) ⓑ \(\sqrt{-36}\)
Try it.
ⓐ \(\text{-}\sqrt{225}\) ⓑ \(\sqrt{-9}\)
Solution
ⓐ \(-15\) ⓑ not real number
Try it.
ⓐ \(\sqrt{-49}\) ⓑ \(\text{-}\sqrt{256}\)
Try it.
ⓐ \(\sqrt[3]{216}\) ⓑ \(\sqrt[4]{256}\)
Solution
ⓐ 6 ⓑ 4
Try it.
ⓐ \(\sqrt[3]{27}\) ⓑ \(\sqrt[4]{16}\) ⓒ \(\sqrt[5]{243}\)
Try it.
ⓐ \(\sqrt[3]{512}\) ⓑ \(\sqrt[4]{81}\) ⓒ \(\sqrt[5]{1}\)
Solution
ⓐ 8 ⓑ 3 ⓒ 1
Try it.
ⓐ \(\sqrt[3]{125}\) ⓑ \(\sqrt[4]{1296}\) ⓒ \(\sqrt[5]{1024}\)
Try it.
ⓐ \(\sqrt[3]{-8}\) ⓑ \(\sqrt[4]{-81}\) ⓒ \(\sqrt[5]{-32}\)
Solution
ⓐ \(-2\) ⓑ \(\text{not real}\) ⓒ \(-2\)
Try it.
ⓐ \(\sqrt[3]{-64}\) ⓑ \(\sqrt[4]{-16}\) ⓒ \(\sqrt[5]{-243}\)
Try it.
ⓐ \(\sqrt[3]{-125}\) ⓑ \(\sqrt[4]{-1296}\) ⓒ \(\sqrt[5]{-1024}\)
Solution
ⓐ \(-5\) ⓑ \(\text{not real}\) ⓒ \(-4\)
Try it.
ⓐ \(\sqrt[3]{-512}\) ⓑ \(\sqrt[4]{-81}\) ⓒ \(\sqrt[5]{-1}\)
Estimate and Approximate Roots
In the following exercises, estimate each root between two consecutive whole numbers.
Try it.
ⓐ \(\sqrt{70}\) ⓑ \(\sqrt[3]{71}\)
Solution
ⓐ \(8<\sqrt{70}<9\)
ⓑ \(4<\sqrt[3]{71}<5\)
Try it.
ⓐ \(\sqrt{55}\) ⓑ \(\sqrt[3]{119}\)
Try it.
ⓐ \(\sqrt{200}\) ⓑ \(\sqrt[3]{137}\)
Solution
ⓐ \(14<\sqrt{200}<15\)
ⓑ \(5<\sqrt[3]{137}<6\)
Try it.
ⓐ \(\sqrt{172}\) ⓑ \(\sqrt[3]{200}\)
In the following exercises, approximate each root and round to two decimal places.
Try it.
ⓐ \(\sqrt{19}\) ⓑ \(\sqrt[3]{89}\) ⓒ \(\sqrt[4]{97}\)
Solution
ⓐ \(\approx 4.36\) ⓑ \(\approx 4.46\)
ⓒ \(\approx 3.14\)
Try it.
ⓐ \(\sqrt{21}\) ⓑ \(\sqrt[3]{93}\) ⓒ \(\sqrt[4]{101}\)
Try it.
ⓐ \(\sqrt{53}\) ⓑ \(\sqrt[3]{147}\) ⓒ \(\sqrt[4]{452}\)
Solution
ⓐ \(\approx 7.28\) ⓑ \(\approx 5.28\)
ⓒ \(\approx 4.61\)
Try it.
ⓐ \(\sqrt{47}\) ⓑ \(\sqrt[3]{163}\) ⓒ \(\sqrt[4]{527}\)
Simplify Variable Expressions with Roots
In the following exercises, simplify using absolute values as necessary.
Try it.
ⓐ \(\sqrt[5]{{u}^{5}}\) ⓑ \(\sqrt[8]{{v}^{8}}\)
Solution
ⓐ u ⓑ \(|v|\)
Try it.
ⓐ \(\sqrt[3]{{a}^{3}}\) ⓑ \(\sqrt[9]{{b}^{9}}\)
Try it.
ⓐ \(\sqrt[4]{{y}^{4}}\) ⓑ \(\sqrt[7]{{m}^{7}}\)
Solution
ⓐ \(|y|\) ⓑ \(m\)
Try it.
ⓐ \(\sqrt[8]{{k}^{8}}\) ⓑ \(\sqrt[6]{{p}^{6}}\)
Try it.
ⓐ \(\sqrt{{x}^{6}}\) ⓑ \(\sqrt{{y}^{16}}\)
Solution
ⓐ \(|{x}^{3}|\) ⓑ \({y}^{8}\)
Try it.
ⓐ \(\sqrt{{a}^{14}}\) ⓑ \(\sqrt{{w}^{24}}\)
Try it.
ⓐ \(\sqrt{{x}^{24}}\) ⓑ \(\sqrt{{y}^{22}}\)
Solution
ⓐ \({x}^{12}\) ⓑ \(|{y}^{11}|\)
Try it.
ⓐ \(\sqrt{{a}^{12}}\) ⓑ \(\sqrt{{b}^{26}}\)
Try it.
ⓐ \(\sqrt[3]{{x}^{9}}\) ⓑ \(\sqrt[4]{{y}^{12}}\)
Solution
ⓐ \({x}^{3}\) ⓑ \(|{y}^{3}|\)
Try it.
ⓐ \(\sqrt[5]{{a}^{10}}\) ⓑ \(\sqrt[3]{{b}^{27}}\)
Try it.
ⓐ \(\sqrt[4]{{m}^{8}}\) ⓑ \(\sqrt[5]{{n}^{20}}\)
Solution
ⓐ \({m}^{2}\) ⓑ \({n}^{4}\)
Try it.
ⓐ \(\sqrt[6]{{r}^{12}}\) ⓑ \(\sqrt[3]{{s}^{30}}\)
Try it.
ⓐ \(\sqrt{49{x}^{2}}\) ⓑ \(\text{-}\sqrt{81{x}^{18}}\)
Solution
ⓐ \(7|x|\) ⓑ \(-9|{x}^{9}|\)
Try it.
ⓐ \(\sqrt{100{y}^{2}}\) ⓑ \(\text{-}\sqrt{100{m}^{32}}\)
Try it.
ⓐ \(\sqrt{121{m}^{20}}\) ⓑ \(\text{-}\sqrt{64{a}^{2}}\)
Solution
ⓐ \(11{m}^{10}\) ⓑ \(-8|a|\)
Try it.
ⓐ \(\sqrt{81{x}^{36}}\) ⓑ \(\text{-}\sqrt{25{x}^{2}}\)
Try it.
ⓐ \(\sqrt[4]{16{x}^{8}}\) ⓑ \(\sqrt[6]{64{y}^{12}}\)
Solution
ⓐ \(2{x}^{2}\) ⓑ \(2{y}^{2}\)
Try it.
ⓐ \(\sqrt[3]{-8{c}^{9}}\) ⓑ \(\sqrt[3]{125{d}^{15}}\)
Try it.
ⓐ \(\sqrt[3]{216{a}^{6}}\) ⓑ \(\sqrt[5]{32{b}^{20}}\)
Solution
ⓐ \(6{a}^{2}\) ⓑ \(2{b}^{4}\)
Try it.
ⓐ \(\sqrt[7]{128{r}^{14}}\) ⓑ \(\sqrt[4]{81{s}^{24}}\)
Try it.
ⓐ \(\sqrt{144{x}^{2}{y}^{2}}\) ⓑ \(\sqrt{169{w}^{8}{y}^{10}}\) ⓒ \(\sqrt[3]{8{a}^{51}{b}^{6}}\)
Solution
ⓐ \(12|xy|\) ⓑ \(13{w}^{4}|{y}^{5}|\)
ⓒ \(2{a}^{17}{b}^{2}\)
Try it.
ⓐ \(\sqrt{196{a}^{2}{b}^{2}}\) ⓑ \(\sqrt{81{p}^{24}{q}^{6}}\) ⓒ \(\sqrt[3]{27{p}^{45}{q}^{9}}\)
Try it.
ⓐ \(\sqrt{121{a}^{2}{b}^{2}}\) ⓑ \(\sqrt{9{c}^{8}{d}^{12}}\) ⓒ \(\sqrt[3]{64{x}^{15}{y}^{66}}\)
Solution
ⓐ \(11|ab|\) ⓑ \(3{c}^{4}{d}^{6}\)
ⓒ \(4{x}^{5}{y}^{22}\)
Try it.
ⓐ \(\sqrt{225{x}^{2}{y}^{2}{z}^{2}}\) ⓑ \(\sqrt{36{r}^{6}{s}^{20}}\) ⓒ \(\sqrt[3]{125{y}^{18}{z}^{27}}\)
Condensed: the full section is in OpenStax Intermediate Algebra 2e.
Now you Không máy tính nào có thể tính toán được nó, nhưng các mảnh của nó có thể tính toán được. Thử một trong những cách dưới đây, hoặc gõ theo ý bạn.
Luyện tập (40)
Đầu tiên thử mỗi câu trên giấy. Cho thấy câu trả lời để kiểm tra; những câu trả lời đã kiểm tra có thể mở trong trình giải cho mỗi bước.
-
Simplify: ⓐ \({(-9)}^{2}\) ⓑ \(\text{-}{9}^{2}\) ⓒ \({(-9)}^{3}.\)
Giải đáp
ⓐ 8; ⓑ −1; ⓒ -729
-
Round \(3.846\) to the nearest hundredth.
Giải đáp
3.85
-
Simplify: ⓐ \({x}^{3}\cdot {x}^{3}\) ⓑ \({y}^{2}\cdot {y}^{2}\cdot {y}^{2}\) ⓒ \({z}^{3}\cdot {z}^{3}\cdot {z}^{3}\cdot {z}^{3}.\)
Giải đáp
ⓐ \({x}^{6}\); ⓑ \({y}^{6}\); ⓒ \({z}^{12}\)
-
Simplify: ⓐ \(\sqrt{144}\) ⓑ \(\text{-}\sqrt{289}.\)
Giải đáp
ⓐ
\(\sqrt{144}\) Since \({12}^{2}=144.\) \(12\)
ⓑ
\(-\sqrt{289}\) Since \({17}^{2}=289\) and the negative is in front of the radical sign. \(-17\) -
Simplify: ⓐ \(\text{-}\sqrt{64}\) ⓑ \(\sqrt{225}.\)
Giải đáp
ⓐ \(-8\) ⓑ 15
-
Simplify: ⓐ \(\sqrt{100}\) ⓑ \(\text{-}\sqrt{121}.\)
Giải đáp
ⓐ 10 ⓑ \(-11\)
-
Simplify: ⓐ \(\sqrt{-196}\) ⓑ \(\text{-}\sqrt{64}.\)
Giải đáp
ⓐ
\(\sqrt{-196}\) There is no real number whose square is \(-196.\) \(\sqrt{-196}\ \text{is not a real number.}\)
ⓑ
\(-\sqrt{64}\) The negative is in front of the radical. \(-8\) -
Simplify: ⓐ \(\sqrt{-169}\) ⓑ \(\text{-}\sqrt{81}.\)
Giải đáp
ⓐ not a real number ⓑ \(-9\)
-
Simplify: ⓐ \(\text{-}\sqrt{49}\) ⓑ \(\sqrt{-121}.\)
Giải đáp
ⓐ \(-7\) ⓑ not a real number
-
Simplify: ⓐ \(\sqrt[3]{64}\) ⓑ \(\sqrt[4]{81}\) ⓒ \(\sqrt[5]{32}.\)
Giải đáp
ⓐ
\(\sqrt[3]{64}\) Since \({4}^{3}=64.\) \(4\)
ⓑ
\(\sqrt[4]{81}\) Since \({(3)}^{4}=81.\) \(3\)
ⓒ
\(\sqrt[5]{32}\) Since \({(2)}^{5}=32.\) \(2\) -
Simplify: ⓐ \(\sqrt[3]{27}\) ⓑ \(\sqrt[4]{256}\) ⓒ \(\sqrt[5]{243}.\)
Giải đáp
ⓐ 3 ⓑ 4 ⓒ 3
-
Simplify: ⓐ \(\sqrt[3]{1000}\) ⓑ \(\sqrt[4]{16}\) ⓒ \(\sqrt[5]{1024}.\)
Giải đáp
ⓐ 10 ⓑ 2 ⓒ 4
-
Simplify: ⓐ \(\sqrt[3]{-125}\) ⓑ \(\sqrt[4]{-16}\) ⓒ \(\sqrt[5]{-243}.\)
Giải đáp
ⓐ
\(\sqrt[3]{-125}\) Since \({(-5)}^{3}=-125.\) \(-5\)
ⓑ
\(\sqrt[4]{-16}\) Think, \({(?)}^{4}=-16.\) No real number raised to the fourth power is negative. Not a real number.
ⓒ
\(\sqrt[5]{-243}\) Since \({(-3)}^{5}=-243.\) \(-3\) -
Simplify: ⓐ \(\sqrt[3]{-27}\) ⓑ \(\sqrt[4]{-256}\) ⓒ \(\sqrt[5]{-32}.\)
Giải đáp
ⓐ \(-3\) ⓑ not real ⓒ \(-2\)
-
Simplify: ⓐ \(\sqrt[3]{-216}\) ⓑ \(\sqrt[4]{-81}\) ⓒ \(\sqrt[5]{-1024}.\)
Giải đáp
ⓐ \(-6\) ⓑ not real ⓒ \(-4\)
-
Estimate each root between two consecutive whole numbers: ⓐ \(\sqrt{105}\) ⓑ \(\sqrt[3]{43}.\)
Giải đáp
ⓐ Think of the perfect square numbers closest to 105. Make a small table of these perfect squares and their squares roots.
Locate 105 between two consecutive perfect squares. \(\sqrt{105}\) is between their square roots. ⓑ Similarly we locate 43 between two perfect cube numbers.
Locate 43 between two consecutive perfect cubes. \(\sqrt[3]{43}\) is between their cube roots. -
Estimate each root between two consecutive whole numbers:
ⓐ \(\sqrt{38}\) ⓑ \(\sqrt[3]{93}\)
Giải đáp
ⓐ \(6<\sqrt{38}<7\)
ⓑ \(4<\sqrt[3]{93}<5\) -
Estimate each root between two consecutive whole numbers:
ⓐ \(\sqrt{84}\) ⓑ \(\sqrt[3]{152}\)
Giải đáp
ⓐ \(9<\sqrt{84}<10\)
ⓑ \(5<\sqrt[3]{152}<6\) -
Round to two decimal places: ⓐ \(\sqrt{17}\) ⓑ \(\sqrt[3]{49}\) ⓒ \(\sqrt[4]{51}.\)
Giải đáp
ⓐ
\(\sqrt{17}\) Use the calculator square root key. \(4.123105626\ldots\) Round to two decimal places. \(4.12\) \(\sqrt{17}\approx 4.12\)
ⓑ
\(\sqrt[3]{49}\) Use the calculator \(\sqrt[y]{x}\) key. \(3.659305710\ldots\) Round to two decimal places. \(3.66\) \(\sqrt[3]{49}\approx 3.66\)
ⓒ
\(\sqrt[4]{51}\) Use the calculator \(\sqrt[y]{x}\) key. \(2.6723451177\ldots\) Round to two decimal places. \(2.67\) \(\sqrt[4]{51}\approx 2.67\) -
Round to two decimal places:
ⓐ \(\sqrt{11}\) ⓑ \(\sqrt[3]{71}\) ⓒ \(\sqrt[4]{127}.\)
Giải đáp
ⓐ \(\approx 3.32\) ⓑ \(\approx 4.14\)
ⓒ \(\approx 3.36\) -
Round to two decimal places:
ⓐ \(\sqrt{13}\) ⓑ \(\sqrt[3]{84}\) ⓒ \(\sqrt[4]{98}.\)
Giải đáp
ⓐ \(\approx 3.61\) ⓑ \(\approx 4.38\)
ⓒ \(\approx 3.15\) -
Simplify: ⓐ \(\sqrt{{x}^{2}}\) ⓑ \(\sqrt[3]{{n}^{3}}\) ⓒ \(\sqrt[4]{{p}^{4}}\) ⓓ \(\sqrt[5]{{y}^{5}}.\)
Giải đáp
ⓐ We use the absolute value to be sure to get the positive root.
\(\sqrt{{x}^{2}}\) Since the index \(n\) is even, \(\sqrt[n]{{a}^{n}}=|a|.\) \(|x|\) ⓑ This is an odd indexed root so there is no need for an absolute value sign.
\(\sqrt[3]{{n}^{3}}\) Since the index \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\) \(n\) ⓒ
\(\sqrt[4]{{p}^{4}}\) Since the index \(n\ \text{is even}\ \sqrt[n]{{a}^{n}}=|a|.\) \(|p|\) ⓓ
\(\sqrt[5]{{y}^{5}}\) Since the index \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\) \(y\) -
Simplify: ⓐ \(\sqrt{{b}^{2}}\) ⓑ \(\sqrt[3]{{w}^{3}}\) ⓒ \(\sqrt[4]{{m}^{4}}\) ⓓ \(\sqrt[5]{{q}^{5}}.\)
Giải đáp
ⓐ \(|b|\) ⓑ w ⓒ \(|m|\) ⓓ q
-
Simplify: ⓐ \(\sqrt{{y}^{2}}\) ⓑ \(\sqrt[3]{{p}^{3}}\) ⓒ \(\sqrt[4]{{z}^{4}}\) ⓓ \(\sqrt[5]{{q}^{5}}.\)
Giải đáp
ⓐ \(|y|\) ⓑ p ⓒ \(|z|\) ⓓ q
-
Simplify: ⓐ \(\sqrt{{x}^{6}}\) ⓑ \(\sqrt{{y}^{16}}.\)
Giải đáp
ⓐ
\(\sqrt{{x}^{6}}\) Since \({({x}^{3})}^{2}={x}^{6}.\) \(\sqrt{{({x}^{3})}^{2}}\) Since the index \(n\) is even \(\sqrt{{a}^{n}}=|a|.\) \(|{x}^{3}|\)
ⓑ
\(\sqrt{{y}^{16}}\) Since \({({y}^{8})}^{2}={y}^{16}.\) \(\sqrt{{({y}^{8})}^{2}}\) Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=|a|.\) \({y}^{8}\) In this case the absolute value sign is not needed as \({y}^{8}\) is positive. -
Simplify: ⓐ \(\sqrt{{y}^{18}}\) ⓑ \(\sqrt{{z}^{12}}.\)
Giải đáp
ⓐ \(|{y}^{9}|\) ⓑ \({z}^{6}\)
-
Simplify: ⓐ \(\sqrt{{m}^{4}}\) ⓑ \(\sqrt{{b}^{10}}.\)
Giải đáp
ⓐ \({m}^{2}\) ⓑ \(|{b}^{5}|\)
-
Simplify: ⓐ \(\sqrt[3]{{y}^{18}}\) ⓑ \(\sqrt[4]{{z}^{8}}.\)
Giải đáp
ⓐ
\(\sqrt[3]{{y}^{18}}\) Since \({({y}^{6})}^{3}={y}^{18}.\) \(\sqrt[3]{{({y}^{6})}^{3}}\) Since \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\) \({y}^{6}\)
ⓑ
\(\sqrt[4]{{z}^{8}}\) Since \({({z}^{2})}^{4}={z}^{8}.\) \(\sqrt[4]{{({z}^{2})}^{4}}\) Since \({z}^{2}\) is positive, we do not need an absolute value sign. \({z}^{2}\) -
Simplify: ⓐ \(\sqrt[4]{{u}^{12}}\) ⓑ \(\sqrt[3]{{v}^{15}}.\)
Giải đáp
ⓐ \(|{u}^{3}|\) ⓑ \({v}^{5}\)
-
Simplify: ⓐ \(\sqrt[5]{{c}^{20}}\) ⓑ \(\sqrt[6]{{d}^{24}}\)
Giải đáp
ⓐ \({c}^{4}\) ⓑ \({d}^{4}\)
-
Simplify: ⓐ \(\sqrt{16{n}^{2}}\) ⓑ \(\text{-}\sqrt{81{c}^{2}}.\)
Giải đáp
ⓐ
\(\sqrt{16{n}^{2}}\) Since \({(4n)}^{2}=16{n}^{2}.\) \(\sqrt{{(4n)}^{2}}\) Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=|a|.\) \(4|n|\)
ⓑ
\(-\sqrt{81{c}^{2}}\) Since \({(9c)}^{2}=81{c}^{2}.\) \(-\sqrt{{(9c)}^{2}}\) Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=|a|.\) \(-9|c|\) -
Simplify: ⓐ \(\sqrt{64{x}^{2}}\) ⓑ \(\text{-}\sqrt{100{p}^{2}}.\)
Giải đáp
ⓐ \(8|x|\) ⓑ \(-10|p|\)
-
Simplify: ⓐ \(\sqrt{169{y}^{2}}\) ⓑ \(\text{-}\sqrt{121{y}^{2}}.\)
Giải đáp
ⓐ \(13|y|\) ⓑ \(-11|y|\)
-
Simplify: ⓐ \(\sqrt[3]{64{p}^{6}}\) ⓑ \(\sqrt[4]{16{q}^{12}}.\)
Giải đáp
ⓐ
\(\sqrt[3]{64{p}^{6}}\) Rewrite \(64{p}^{6}\) as \({(4{p}^{2})}^{3}.\) \(\sqrt[3]{{(4{p}^{2})}^{3}}\) Take the cube root. \(4{p}^{2}\)
ⓑ
\(\sqrt[4]{16{q}^{12}}\) Rewrite the radicand as a fourth power. \(\sqrt[4]{{(2{q}^{3})}^{4}}\) Take the fourth root. \(2|{q}^{3}|\) -
Simplify: ⓐ \(\sqrt[3]{27{x}^{27}}\) ⓑ \(\sqrt[4]{81{q}^{28}}.\)
Giải đáp
ⓐ \(3{x}^{9}\) ⓑ \(3|{q}^{7}|\)
-
Simplify: ⓐ \(\sqrt[3]{125{q}^{9}}\) ⓑ \(\sqrt[5]{243{q}^{25}}.\)
Giải đáp
ⓐ \(5{q}^{3}\) ⓑ \(3{q}^{5}\)
-
Simplify: ⓐ \(\sqrt{36{x}^{2}{y}^{2}}\) ⓑ \(\sqrt{121{a}^{6}{b}^{8}}\) ⓒ \(\sqrt[3]{64{p}^{63}{q}^{9}}.\)
Giải đáp
ⓐ
\(\sqrt{36{x}^{2}{y}^{2}}\) Since \({(6xy)}^{2}=36{x}^{2}{y}^{2}\) \(\sqrt{{(6xy)}^{2}}\) Take the square root. \(6|xy|\)
ⓑ
\(\sqrt{121{a}^{6}{b}^{8}}\) Since \({(11{a}^{3}{b}^{4})}^{2}=121{a}^{6}{b}^{8}\) \(\sqrt{{(11{a}^{3}{b}^{4})}^{2}}\) Take the square root. \(11|{a}^{3}|{b}^{4}\)
ⓒ
\(\sqrt[3]{64{p}^{63}{q}^{9}}\) Since \({(4{p}^{21}{q}^{3})}^{3}=64{p}^{63}{q}^{9}\) \(\sqrt[3]{{(4{p}^{21}{q}^{3})}^{3}}\) Take the cube root. \(4{p}^{21}{q}^{3}\) -
Simplify: ⓐ \(\sqrt{100{a}^{2}{b}^{2}}\) ⓑ \(\sqrt{144{p}^{12}{q}^{20}}\) ⓒ \(\sqrt[3]{8{x}^{30}{y}^{12}}\)
Giải đáp
ⓐ \(10|ab|\) ⓑ \(12{p}^{6}{q}^{10}\)
ⓒ \(2{x}^{10}{y}^{4}\) -
Simplify: ⓐ \(\sqrt{225{m}^{2}{n}^{2}}\) ⓑ \(\sqrt{169{x}^{10}{y}^{14}}\) ⓒ \(\sqrt[3]{27{w}^{36}{z}^{15}}\)
Giải đáp
ⓐ \(15|mn|\) ⓑ \(13|{x}^{5}{y}^{7}|\)
ⓒ \(3{w}^{12}{z}^{5}\) -
ⓐ \(\sqrt{64}\) ⓑ \(\text{-}\sqrt{81}\)
Giải đáp
ⓐ 8 ⓑ \(-9\)
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Làm thế nào: Simplify Expressions with Roots
- Simplify expressions with roots
- Estimate and approximate roots
- Simplify variable expressions with roots
- If
- The square root of
- If
- The principal
- When
Câu hỏi người ta hỏi
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.
Một số phần của trang này được chuyển từ OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Được nén lại và giải thích lại ở đây; lỗi là của chúng tôi. Bởi vì nguồn của nó được cấp phép cho sử dụng không thương mại, bài học này là miễn phí, không bán gì, và được chia sẻ dưới cùng một giấy phép.
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