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We can continue this way to see such roots as fourth roots, fifth roots, sixth roots, and so on.
There is a symbol used to indicate roots of a number. It is called the radical sign $\sqrt[\mathrm{n}]{\phantom{\rule{16px}{0ex}}}$
We discuss particular roots using the radical sign as follows:
$\sqrt{\text{49}}$ = 7 since $7\cdot 7={7}^{2}=\text{49}$
$\sqrt[3]{8}=2$ since $2\cdot 2\cdot 2={2}^{3}=8$
$\sqrt[4]{\text{81}}=3$ since $3\cdot 3\cdot 3\cdot 3={3}^{4}=\text{81}$
In an expression such as $\sqrt[5]{\text{32}}$
Find each root.
$\sqrt{\text{25}}$ To determine the square root of 25, we ask, "What whole number squared equals 25?" From our experience with multiplication, we know this number to be 5. Thus,
$\sqrt{\text{25}}=5$
Check: $5\cdot 5={5}^{2}=\text{25}$
$\sqrt[5]{\text{32}}$ To determine the fifth root of 32, we ask, "What whole number raised to the fifth power equals 32?" This number is 2.
$\sqrt[5]{\text{32}}=2$
Check: $2\cdot 2\cdot 2\cdot 2\cdot 2={2}^{5}=\text{32}$
Find the following roots using only a knowledge of multiplication.
Calculators with the $\sqrt{x}$ , ${y}^{x}$ , and $1/x$ keys can be used to find or approximate roots.
Use the calculator to find $\sqrt{\text{121}}$
Display Reads | ||
Type | 121 | 121 |
Press | $\sqrt{x}$ | 11 |
Find $\sqrt[7]{\text{2187}}$ .
Display Reads | ||
Type | 2187 | 2187 |
Press | ${y}^{x}$ | 2187 |
Type | 7 | 7 |
Press | $1/x$ | .14285714 |
Press | = | 3 |
$\sqrt[7]{\text{2187}}=3$ (Which means that ${3}^{7}=2187$ .)
Use a calculator to find the following roots.
For the following problems, write the expressions using exponential notation.
$\text{12}\cdot \text{12}$
$\text{10}\cdot \text{10}\cdot \text{10}\cdot \text{10}\cdot \text{10}\cdot \text{10}$
$\text{826}\cdot \text{826}\cdot \text{826}$
${\text{826}}^{3}$
$\mathrm{3,}\text{021}\cdot \mathrm{3,}\text{021}\cdot \mathrm{3,}\text{021}\cdot \mathrm{3,}\text{021}\cdot \mathrm{3,}\text{021}$
$\underset{\text{85 factors of 6}}{\underbrace{6\xb76\xb7\xb7\xb7\xb7\xb76}}$
${6}^{\text{85}}$
$\underset{\text{112 factors of 2}}{\underbrace{2\xb72\xb7\xb7\xb7\xb7\xb72}}$
$\underset{\text{3,008 factors of 1}}{\underbrace{1\xb71\xb7\xb7\xb7\xb7\xb71}}$
${1}^{\text{3008}}$
For the following problems, expand the terms. (Do not find the actual value.)
${5}^{3}$
${\text{15}}^{2}$
${\text{117}}^{5}$
$\text{117}\cdot \text{117}\cdot \text{117}\cdot \text{117}\cdot \text{117}$
${\text{61}}^{6}$
For the following problems, determine the value of each of the powers. Use a calculator to check each result.
${3}^{2}$
${1}^{2}$
${\text{11}}^{2}$
${\text{13}}^{2}$
${1}^{4}$
${7}^{3}$
${\text{10}}^{3}$
$\text{10}\cdot \text{10}\cdot \text{10}=\mathrm{1,}\text{000}$
${\text{100}}^{2}$
${5}^{5}$
${6}^{2}$
${1}^{\text{28}}$
${2}^{7}$
$2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=\text{128}$
${0}^{5}$
${5}^{8}$
${6}^{9}$
$6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6=\text{10},\text{077},\text{696}$
${\text{25}}^{3}$
${\text{42}}^{2}$
$\text{42}\cdot \text{42}=\mathrm{1,}\text{764}$
${\text{31}}^{3}$
${\text{15}}^{5}$
$\text{15}\cdot \text{15}\cdot \text{15}\cdot \text{15}\cdot \text{15}=\text{759},\text{375}$
${2}^{\text{20}}$
${\text{816}}^{2}$
$\text{816}\cdot \text{816}=\text{665},\text{856}$
For the following problems, find the roots (using your knowledge of multiplication). Use a calculator to check each result.
$\sqrt{9}$
$\sqrt{\text{36}}$
$\sqrt{\text{121}}$
$\sqrt{\text{169}}$
$\sqrt[3]{\text{27}}$
$\sqrt[4]{\text{256}}$
$\sqrt[7]{1}$
$\sqrt{\text{900}}$
$\sqrt{\text{324}}$
For the following problems, use a calculator with the keys $\sqrt{x}$ , ${y}^{x}$ , and $1/x$ to find each of the values.
$\sqrt{\text{676}}$
$\sqrt{\text{46},\text{225}}$
$\sqrt[3]{\mathrm{3,}\text{375}}$
$\sqrt[8]{\mathrm{5,}\text{764},\text{801}}$
$\sqrt[8]{\text{16},\text{777},\text{216}}$
$\sqrt[4]{\text{160},\text{000}}$
( [link] ) Use the numbers 3, 8, and 9 to illustrate the associative property of addition.
( [link] ) In the multiplication $8\cdot 4=\text{32}$ , specify the name given to the numbers 8 and 4.
8 is the multiplier; 4 is the multiplicand
( [link] ) Does the quotient $\text{15}\xf70$ exist? If so, what is it?
( [link] ) Does the quotient $0\xf7\text{15}$ exist? If so, what is it?
Yes; 0
( [link] ) Use the numbers 4 and 7 to illustrate the commutative property of multiplication.
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