9.1E: Exercises

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Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

Example $\PageIndex{34}$

$\sqrt{36}$

Answer: 6

Example $\PageIndex{35}$

$\sqrt{4}$

Example $\PageIndex{36}$

$\sqrt{64}$

Answer: 8

Example $\PageIndex{37}$

$\sqrt{169}$

Example $\PageIndex{38}$

$\sqrt{9}$

Answer: 3

Example $\PageIndex{39}$

$\sqrt{16}$

Example $\PageIndex{40}$

$\sqrt{100}$

Answer: 10

Example $\PageIndex{41}$

$\sqrt{144}$

Example $\PageIndex{42}$

$−\sqrt{4}$

Answer: −2

Example $\PageIndex{43}$

$−\sqrt{100}$

Example $\PageIndex{44}$

$−\sqrt{1}$

Answer: −1

Example $\PageIndex{45}$

$−\sqrt{121}$

Example $\PageIndex{46}$

$\sqrt{−121}$

Answer: not a real number

Example $\PageIndex{47}$

$\sqrt{−36}$

Example $\PageIndex{48}$

$\sqrt{−9}$

Answer: not a real number

Example $\PageIndex{49}$

$\sqrt{−49}$

Example $\PageIndex{50}$

$\sqrt{9+16}$

Answer: 5

Example $\PageIndex{51}$

$\sqrt{25+144}$

Example $\PageIndex{52}$

$\sqrt{9}+\sqrt{16}$

Answer: 7

Example $\PageIndex{53}$

$\sqrt{25}+\sqrt{144}$

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

Example $\PageIndex{54}$

$\sqrt{70}$

Answer: $8<\sqrt{70}<9$

Example $\PageIndex{55}$

$\sqrt{55}$

Example $\PageIndex{56}$

$\sqrt{200}$

Answer: $14<\sqrt{200}<15$

Example $\PageIndex{57}$

$\sqrt{172}$

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

Example $\PageIndex{58}$

$\sqrt{19}$

Answer: 4.36

Example $\PageIndex{59}$

$\sqrt{21}$

Example $\PageIndex{60}$

$\sqrt{53}$

Answer: 7.28

Example $\PageIndex{61}$

$\sqrt{47}$

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Example $\PageIndex{62}$

$\sqrt{y^2}$

Answer: y

Example $\PageIndex{63}$

$\sqrt{b^2}$

Example $\PageIndex{64}$

$\sqrt{a^{14}}$

Answer: $a^7$

Example $\PageIndex{65}$

$\sqrt{w^{24}}$

Example $\PageIndex{66}$

$\sqrt{49x^2}$

Answer: $7x$

Example $\PageIndex{67}$

$\sqrt{100y^2}$

Example $\PageIndex{68}$

$\sqrt{121m^{20}}$

Answer: $11m^{10}$

Example $\PageIndex{69}$

$25h^{44}$

Example $\PageIndex{70}$

$\sqrt{81x^{36}}$

Answer: $9x^{18}$

Example $\PageIndex{71}$

$\sqrt{144z^{84}}$

Example $\PageIndex{72}$

$−\sqrt{81x^{18}}$

Answer: $−9x^9$

Example $\PageIndex{73}$

$−\sqrt{100m^{32}}$

Example $\PageIndex{74}$

$−\sqrt{64a^2}$

Answer: $−8a$

Example $\PageIndex{75}$

$−\sqrt{25x^2}$

Example $\PageIndex{76}$

$\sqrt{144x^{2}y^{2}}$

Answer: $12xy$

Example $\PageIndex{77}$

$\sqrt{196a^{2}b^{2}}$

Example $\PageIndex{78}$

$\sqrt{169w^{8}y^{10}}$

Answer: $13w^{4}y^{5}$

Example $\PageIndex{79}$

$\sqrt{81p^{24}q^{6}}$

Example $\PageIndex{80}$

$\sqrt{9c^{8}d^{12}}$

Answer: $3c^{4}d^{6}$

Examlpe $\PageIndex{81}$

$\sqrt{36r^{6}s^{20}}$

Everyday Math

Example $\PageIndex{82}$

Decorating Denise wants to have a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be?

Answer: 25 centimeters

Example $\PageIndex{83}$

Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for 2025 square inch tiles. How long can a side of the mosaic be?

Writing Exercises

Example $\PageIndex{84}$

Why is there no real number equal to $\sqrt{−64}$?

Answer: Answers will vary.

Example $\PageIndex{85}$

What is the difference between $9^{2}$ and $\sqrt{9}$?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

$This table has four columns and five rows. The columns are labeled, “I can…,” “Confidentally,” “With some help,” and “No – I don’t get it!” Under the “I can…,” column are, “simplify expressions with square roots.,” “estimate square roots.,” “approximate square roots.,” and “4) simplify variable expressions with square roots.” All the other rows under the different columns are empty.$

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Practice Makes Perfect

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Example \(\PageIndex{80}\)

Examlpe \(\PageIndex{81}\)

Everyday Math

Example \(\PageIndex{82}\)

Example \(\PageIndex{83}\)

Writing Exercises

Example \(\PageIndex{84}\)

Example \(\PageIndex{85}\)

Self Check