5.7 Simplify and Use Square Roots

Before you get started, take this readiness quiz.

Simplify: $\left(−9\right)^{2} .$
If you missed this problem, review the reference.

Round $3.846$ to the nearest hundredth.
If you missed this problem, review the reference.

Evaluate $12 d$ for $d = 80 .$
If you missed this problem, review the reference.

Simplify: ⓐ $\sqrt{25}$ⓑ $\sqrt{121} .$

Simplify: ⓐ $\sqrt{36}$ⓑ $\sqrt{169} .$

Simplify: ⓐ $\sqrt{16}$ⓑ $\sqrt{196} .$

Simplify. ⓐ $- \sqrt{9}$ⓑ $- \sqrt{144 .}$

Simplify: ⓐ $- \sqrt{4}$ⓑ $- \sqrt{225} .$

Simplify: ⓐ $- \sqrt{81}$ⓑ $- \sqrt{64} .$

Simplify: ⓐ $\sqrt{−169}$ⓑ $- \sqrt{121} .$

Simplify: ⓐ $\sqrt{−196}$ⓑ $- \sqrt{81} .$

Simplify: ⓐ $\sqrt{−49}$ⓑ $- \sqrt{121} .$

Simplify: ⓐ $\sqrt{25} + \sqrt{144}$ⓑ $\sqrt{25 + 144} .$

Simplify: ⓐ $\sqrt{9} + \sqrt{16}$ⓑ $\sqrt{9 + 16} .$

Simplify: ⓐ $\sqrt{64 + 225}$ⓑ $\sqrt{64} + \sqrt{225} .$

Estimate $\sqrt{38}$ between two consecutive whole numbers.

Estimate $\sqrt{84}$ between two consecutive whole numbers.

Round $\sqrt{17}$ to two decimal places using a calculator.

Round $\sqrt{11}$ to two decimal places.

Round $\sqrt{13}$ to two decimal places.

Simplify: $\sqrt{x^{2}} .$

Simplify: $\sqrt{y^{2}} .$

Simplify: $\sqrt{m^{2}} .$

Simplify: $\sqrt{16 x^{2}} .$

Simplify: $\sqrt{64 x^{2}} .$

Simplify: $\sqrt{169 y^{2}} .$

Simplify: $- \sqrt{81 y^{2}} .$

Simplify: $- \sqrt{121 y^{2}} .$

Simplify: $- \sqrt{100 p^{2}} .$

Simplify: $\sqrt{36 x^{2} y^{2}} .$

Simplify: $\sqrt{100 a^{2} b^{2}} .$

Simplify: $\sqrt{225 m^{2} n^{2}} .$

Mike and Lychelle want to make a square patio. They have enough concrete for an area of $200$ square feet. To the nearest tenth of a foot, how long can a side of their square patio be?

Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of $370$ square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?

Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of $2704$ square centimeters. How long can a side of his mosaic be?

Christy dropped her sunglasses from a bridge $400$ feet above a river. How many seconds does it take for the sunglasses to reach the river?

A helicopter drops a rescue package from a height of $1296$ feet. How many seconds does it take for the package to reach the ground?

A window washer drops a squeegee from a platform $196$ feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?

After a car accident, the skid marks for one car measured $190$ feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?

An accident investigator measured the skid marks of a car and found their length was $76$ feet. To the nearest tenth, what was the speed of the car before the brakes were applied?

The skid marks of a vehicle involved in an accident were $122$ feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?

Simplify.

$\sqrt{36}$

Simplify.

$\sqrt{64}$

Simplify.

$- \sqrt{4}$

Simplify.

$- \sqrt{1}$

Simplify.

$\sqrt{−121}$

Simplify.

$\sqrt{−9}$

Simplify.

$\sqrt{9 + 16}$

Simplify.

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

Estimate each square root between two consecutive whole numbers.

$\sqrt{70}$

Estimate each square root between two consecutive whole numbers.

$\sqrt{200}$

Use a calculator to approximate each square root and round to two decimal places.

$\sqrt{19}$

Use a calculator to approximate each square root and round to two decimal places.

$\sqrt{53}$

Simplify. (Assume all variables are greater than or equal to zero.)

$\sqrt{y^{2}}$

Simplify. (Assume all variables are greater than or equal to zero.)

$\sqrt{49 x^{2}}$

Simplify. (Assume all variables are greater than or equal to zero.)

$- \sqrt{64 a^{2}}$

Simplify. (Assume all variables are greater than or equal to zero.)

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

Solve. Round to one decimal place.

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of $75$ square feet. How long can a side of his garden be?

Solve. Round to one decimal place.

Gravity An airplane dropped a flare from a height of $1,024$ feet above a lake. How many seconds did it take for the flare to reach the water?

Solve. Round to one decimal place.

Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, $4,000$ feet above the Colorado River. How many seconds did it take for the hammer to reach the river?

Solve. Round to one decimal place.

Accident investigation The skid marks from a car involved in an accident measured $216$ feet. What was the speed of the car before the brakes were applied?

Solve. Round to one decimal place.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was $117$ feet. What was the speed of the vehicle before the brakes were applied?

Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for $2,025$ tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?

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

Name each decimal.

$0.375$

Name each decimal.

$5.24$

Name each decimal.

$−4.09$

Write as a decimal.

nine hundredths

Write as a decimal.

ten and thirty-five thousandths

Write as a decimal.

negative five hundredths

Convert each decimal to a fraction. Simplify the answer if possible.

$0.825$

Convert each decimal to a fraction. Simplify the answer if possible.

$3.64$

Order each of the following pairs of numbers, using $<$ or $>.$

$0.6 ___ 0.8$

Order each of the following pairs of numbers, using $<$ or $>.$

$0.803 ____ 0.83$

Round each number to the nearest: ⓐ hundredth ⓑ tenth ⓒ whole number.

$12.529$

Round each number to the nearest: ⓐ hundredth ⓑ tenth ⓒ whole number.

$5.897$

Add or subtract.

$32.89 - 8.22$

Add or subtract.

$10.2 + 14.631$

Add or subtract.

$1.83 - 4.2$

Multiply.

$\left(−6.4\right) \left(0.25\right)$

Multiply.

$\left(15.4\right) \left(1000\right)$

Divide.

$4.32 \div 24$

Divide.

$\left(−0.8\right) \div \left(−0.2\right)$

Divide.

$9 \div 0.045$

Use the strategy for applications to solve.

Jessie put $8$ gallons of gas in her car. One gallon of gas costs $\$\text{3}.\text{528}.$ How much did Jessie owe for all the gas?

Use the strategy for applications to solve.

Alice bought a roll of paper towels that cost $\$\text{2}.\text{49}.$ She had a coupon for $\$\text{0}.\text{35}$ off, and the store doubled the coupon. How much did Alice pay for the paper towels?

Convert each fraction to a decimal.

$\frac{7}{8}$

Convert each fraction to a decimal.

$- \frac{21}{4}$

Convert each fraction to a decimal.

$\frac{6}{11}$

Order each pair of numbers, using $<$ or $>.$

$\frac{3}{5} ___ 0 .$

Order each pair of numbers, using $<$ or $>.$

$- \frac{5}{12} ___ −0.42$

Order each pair of numbers, using $<$ or $>.$

$0.33 ___ \frac{5}{16}$

Write each set of numbers in order from least to greatest.

$\frac{7}{9} , 0.75 , \frac{11}{15}$

Simplify

$\frac{3}{4} \left(15.44 - 7.4\right)$

Simplify

$1.6 + \frac{3}{8}$

Simplify

$- \frac{2}{5} \cdot \frac{9}{10} + 0.14$

Approximate the ⓐ circumference and ⓑ area of each circle.

$\text{radius} = \text{3}.\text{5 ft}.$

Approximate the ⓐ circumference and ⓑ area of each circle.

$\text{diameter} = \text{11 cm}$

Determine whether the each number is a solution of the given equation.

$y + 3.2 = −1.5$
ⓐ $y = 1.7$ⓑ $y = −4.7$

Determine whether the each number is a solution of the given equation.

$0.45 v = −40.5$
ⓐ $v = −18.225$ⓑ $v = −90$

Solve.

$h + 5.91 = 2.4$

Solve.

$p - 4.3 = −1.65$

Solve.

$j - 7.42 = −3.7$

Solve.

$−8.6 x = 34.4$

Solve.

$\frac{a}{0.3} = −24$

Solve.

$\frac{s}{−2.5} = −10$

Translate and solve.

The product of $−5.9$ and $x$ is $−3.54 .$

Translate and solve.

The sum of $m$ and $\left(−4.03\right)$ is $6.8 .$

Find the mean of the numbers.

$\$\text{270}$, $\$\text{310}.\text{50}$, $\$\text{243}.\text{75}$, and$\$\text{252}.\text{15}$

Find the mean of the numbers.

In the last three months, Raul’s water bills were $\$\text{31}.\text{45} , \$\text{48}.\text{76} , \text{and} \$\text{42}.\text{60} .$ Find the mean.

Find the median.

$25$, $23$, $24$, $26$, $29$, $19$, $18$, $32$

Find the median.

The number of clients at Miranda’s beauty salon each weekday last week were $18 , 7 , 12 , 16 , \text{and} 20 .$ Find the median number of clients.

Identify the mode of the numbers.

The number of siblings of a group of students: $2$, $0$, $3$, $2$, $4$, $1$, $6$, $5$, $4$, $1$, $2$, $3$

Solve. (Round decimals to three places.)

Luc has to read $3$ novels and $12$ short stories for his literature class. The professor will choose one reading at random for the final exam. Find the probability that the professor will choose a novel for the final exam. Express your answer as a fraction and as a decimal.

Write each ratio as a fraction. Simplify the answer if possible.

$56$ to $32$

Write each ratio as a fraction. Simplify the answer if possible.

$1.2$ to $1.8$

Write each ratio as a fraction. Simplify the answer if possible.

$2 \frac{1}{3} \text{to} 5 \frac{1}{4}$

Write each ratio as a fraction. Simplify the answer if possible.

$28$ inches to $3$ feet

Write each rate as a fraction. Simplify the answer if possible.

$90$ pounds per $7.5$ square inches

Write each rate as a fraction. Simplify the answer if possible.

$\$\text{612}.\text{50}$ for $35$ hours

Find the unit rate.

$90$ pounds per $7.5$ square inches

Find the unit rate.

$\$\text{612}.\text{50}$ for $35$ hours

Find the unit price.

Highlighters: $6$ for $\$\text{2}.\text{52}$

Find the unit price.

Anna bought a pack of $8$ kitchen towels for $\$\text{13}.\text{20} .$ How much did each towel cost? Round to the nearest cent if necessary.

Find each unit price and then determine the better buy.

Vitamins: $60$ tablets for $\$\text{6}.\text{49}$ or $100$ for $\$\text{11}.\text{99} ?$

Translate the English phrase into an algebraic expression.

$a$ adults to $45$ children

Translate the English phrase into an algebraic expression.

the ratio of $19$ and the sum of $3$ and $n$

Simplify.

$\sqrt{144}$

Simplify.

$- \sqrt{81}$

Simplify.

$\sqrt{−36}$

Simplify.

$\sqrt{64 + 225}$

Estimate each square root between two consecutive whole numbers.

$\sqrt{155}$

Approximate each square root and round to two decimal places.

$\sqrt{57}$

Simplify. (Assume all variables are greater than or equal to zero.)

$\sqrt{64 b^{2}}$

Simplify. (Assume all variables are greater than or equal to zero.)

$\sqrt{225 m^{2} n^{2}}$

Simplify. (Assume all variables are greater than or equal to zero.)

$\sqrt{49 y^{2}}$

Simplify. (Assume all variables are greater than or equal to zero.)

$\sqrt{121 c^{2} d^{2}}$

Solve. Round to one decimal place.

Landscaping Janet wants to plant a square flower garden in her yard. She has enough topsoil to cover an area of $30$ square feet. How long can a side of the flower garden be?

Solve. Round to one decimal place.

Accident investigation The skid marks of a car involved in an accident were $216$ feet. How fast had the car been going before applying the brakes?

Write $1.73$ as a fraction.

Round $16.749$ to the nearest ⓐ tenth ⓑ hundredth ⓒ whole number

Simplify each expression.

$15.4 + 3.02$

Simplify each expression.

$\left(0.64\right) \left(0.3\right)$

Simplify each expression.

$0.96 \div \left(−12\right)$

Simplify each expression.

$−0.6 \div \left(−0.3\right)$

Simplify each expression.

$24 \div \left(0.1 + 0.02\right)$

Simplify each expression.

$1.6 + \frac{3}{8}$

Solve.

$m + 3.7 = 2.5$

Solve.

$−6.5 y = −57.2$

Solve.

Three friends went out to dinner and agreed to split the bill evenly. The bill was $\$\text{79}.\text{35} .$ How much should each person pay?

Solve.

The ages, in months, of $10$ children in a preschool class are:
$55$, $55$, $50$, $51$, $52$, $50$, $53$, $51$, $55$, $49$
Find the ⓐ mean ⓑ median ⓒ mode

Solve.

Find each unit price and then the better buy.
Laundry detergent: $64$ ounces for $\$\text{10}.\text{99}$ or $48$ ounces for $\$\text{8}.\text{49}$

Simplify.

$\sqrt{144 n^{2}}$

Simplify.

Yanet wants a square patio in her backyard. She has $225$ square feet of tile. How long can a side of the patio be?