Roots

You already know that the square root of 99 is

9=3\sqrt{9}=3

because 32=93^2=9. Also, you might know that the third root of 88 is

83=2\sqrt[3]{8}=2

because 23=82^3=8. It is straightforward to define the nn-root of a number.

Definition 1

Consider a number aa and a natural number nn. The n-th root of aa, denoted by

an\sqrt[n]{a}

is any positive number bb with bn=ab^n=a.

The symbol a\sqrt{\text{\phantom{a}}} is called the radical, and the number aa is the radicand.

For example, it is

2435=3\sqrt[5]{243}=3

because 35=2433^5=243. Here are some exercises ... . You can use the calculator to verify your results, the symbol to use is

  □\sqrt[\square]{\,\,}

which you reach with the keys

2nd and x□\boxed{2nd} \text{ and } \boxed{x^\square}
Exercise 1

  1. Without calculator, determine the value of the following roots. You should verify your result using the calculator.

    1. 625\sqrt{625}
    2. 1\sqrt{1}
    3. −1\sqrt{-1}
    4. 03\sqrt[3]{0}
    5. 13\sqrt[3]{1}
    6. −13\sqrt[3]{-1}
    7. 466\sqrt[6]{4^6}
    8. 1232\sqrt{123^2}
    9. 1253\sqrt[3]{125}
    10. 273\sqrt[3]{27}
    11. 409612\sqrt[12]{4096}
    12. 325\sqrt[5]{32}
    13. −83\sqrt[3]{-8}
    14. 814\sqrt[4]{81}
    15. 100 000 0004\sqrt[4]{100\,000\,000}
    16. 125 0003\sqrt[3]{125\,000}
    17. 14\sqrt{\frac{1}{4}}
    18. 1533\sqrt[3]{\frac{1}{5^3}}
    19. 0.01\sqrt{0.01}
    20. 0.25\sqrt{0.25}
    21. 183\sqrt[3]{\frac{1}{8}}
    22. 1814\sqrt[4]{\frac{1}{81}}
    23. 12583\sqrt[3]{\frac{125}{8}}
    24. 11287\sqrt[7]{\frac{1}{128}}
    25. 62512964\sqrt[4]{\frac{625}{1296}}
    26. 23⋅23⋅233\sqrt[3]{2^3\cdot 2^3 \cdot 2^3}
    27. 3433\sqrt[3]{343}
    28. 416\sqrt{\frac{4}{16}}
    29. 8273\sqrt[3]{\frac{8}{27}}
    30. 812564\sqrt[4]{\frac{81}{256}}
    31. 25+144\sqrt{25+144}
    32. 32+42\sqrt{3^2+4^2}
    33. 54+6835\sqrt{4}+6\sqrt[3]{8}

  2. Argue, why are the following equations correct for every positive value xx?

    1. x2=x\sqrt{x^2}=x
    2. x33=x\sqrt[3]{x^3}=x
    3. xnn=x\sqrt[n]{x^n}=x for all natural numbers nn?

  3. Without using the calculator, find positive values for xx such that

    1. x2=576x^2=576

    2. x2−0.64=0x^2-0.64=0

    3. x5=210x^5=2^{10}

    4. x3−116=0x^3-11^6=0

    5. 3x5−96=03x^5-96=0

Solution

  1. The positive values are

    1. 625=25\sqrt{625}=25
    2. 1=1\sqrt{1}=1
    3. −1=\sqrt{-1}= does not exist
    4. 03=0\sqrt[3]{0}=0
    5. 13=1\sqrt[3]{1}=1
    6. −13=−1\sqrt[3]{-1}=-1
    7. 466=4\sqrt[6]{4^6}=4
    8. 1232=123\sqrt{123^2}=123
    9. 1253=5\sqrt[3]{125}=5
    10. 273=3\sqrt[3]{27}=3
    11. 409612=2\sqrt[12]{4096}=2
    12. 325=2\sqrt[5]{32}=2
    13. −83=−2\sqrt[3]{-8}=-2
    14. 814=3\sqrt[4]{81}=3
    15. 100 000 0004=100\sqrt[4]{100\,000\,000}=100
    16. 125 0003=50\sqrt[3]{125\,000}=50
    17. 14=12\sqrt{\frac{1}{4}}=\frac{1}{2}
    18. 1533=15=0.2\sqrt[3]{\frac{1}{5^3}}=\frac{1}{5}=0.2
    19. 0.01=0.1\sqrt{0.01}=0.1
    20. 0.25=0.5\sqrt{0.25}=0.5
    21. 183=12\sqrt[3]{\frac{1}{8}}=\frac{1}{2}
    22. 1814=13\sqrt[4]{\frac{1}{81}}=\frac{1}{3}
    23. 12583=52=2.5\sqrt[3]{\frac{125}{8}}=\frac{5}{2}=2.5
    24. 11287=12\sqrt[7]{\frac{1}{128}}=\frac{1}{2}
    25. 62512964=56\sqrt[4]{\frac{625}{1296}}=\frac{5}{6}
    26. 23⋅23⋅233=23\sqrt[3]{2^3\cdot 2^3 \cdot 2^3}=2^3
    27. 3433=7\sqrt[3]{343}=7
    28. 416=24=12\sqrt{\frac{4}{16}}=\frac{2}{4}=\frac{1}{2}
    29. 8273=23\sqrt[3]{\frac{8}{27}}=\frac{2}{3}
    30. 812564=34\sqrt[4]{\frac{81}{256}}=\frac{3}{4}
    31. 25+144=13\sqrt{25+144}=13
    32. 32+42=5\sqrt{3^2+4^2}=5
    33. 54+683=5⋅2+6⋅2=225\sqrt{4}+6\sqrt[3]{8}=5\cdot 2+6\cdot 2=22

  2. Argue, why are the following equations correct for every positive value xx?

    1. x2=x\sqrt{x^2}=x, because x2x^2 is the radicand
    2. x33=x\sqrt[3]{x^3}=x, because x3x^3 is the radicand
    3. xnn=x\sqrt[n]{x^n}=x because xnx^n is the radicand

  3. Bring the xx on one side, the number on the other side, and then apply the root on both sides. We get

    1. x2=576→x2⏟=x=576=24x^2=576 \rightarrow \underbrace{\sqrt{x^2}}_{=x}=\sqrt{576}=24
    2. x2=0.64→x2⏟=x=0.64=0.8x^2=0.64\rightarrow \underbrace{\sqrt{x^2}}_{=x}=\sqrt{0.64}=0.8
    3. x5=210→x55⏟=x=2105=(22)55=22=4x^5=2^{10}\rightarrow \underbrace{\sqrt[5]{x^5}}_{=x}=\sqrt[5]{2^{10}}=\sqrt[5]{(2^2)^{5}}=2^2=4
    4. x3=116→x33⏟=x=1163=(112)33=112=121x^3=11^6\rightarrow \underbrace{\sqrt[3]{x^3}}_{=x}=\sqrt[3]{11^6}=\sqrt[3]{(11^2)^{3}}=11^2=121
    5. x5=32→x55⏟=x=325=2x^5=32\rightarrow \underbrace{\sqrt[5]{x^5}}_{=x}=\sqrt[5]{32}=2