Weekend Bike Repair: Root Checks

Use square and cube roots to check workshop plans.

Bench Checks

Maya and Luis run a weekend bike-repair business. Complete their math checks without a calculator. The symbol √ means the nonnegative square root.

Evaluate each root.

√169 = ∛216 =

Which statement correctly evaluates −√100?

A. It has no real value.B. Its values are −10 and 10.C. Its value is −10.D. Its value is 10.

∛(−1000) =

Which choice lists all real solutions of x² = 49/64?

A. x = −7/32 or x = 7/32B. x = −49/64 or x = 49/64C. x = 7/8 onlyD. x = −7/8 or x = 7/8

Solve x³ = 1/8. Write your answer as a fraction.

x =

Equation Checks

Pay attention to whether an equation contains a square or a cube. Include every real solution when asked.

Solve x² = 225.

Negative solution: x =
Positive solution: x =

Which choice lists all real solutions of x³ = −729?

A. x = −9 onlyB. x = −9 or x = 9C. x = −243 onlyD. x = 9 only

Maya compares √64 and ∛64. Which statement is correct?

A. √64 is 8, and ∛64 is 4.B. √64 is 8, and ∛64 is 8.C. √64 is 4, and ∛64 is 8.D. √64 is 32, and ∛64 is 16.

Select every equation that has exactly two real solutions.

Workshop Decisions

Apply roots to the repair business. Use positive values for physical lengths.

10.

Luis cuts a square protective mat with an area of 196 square inches. If its side length is x inches, then x² = 196.

How long is each side? inches

11.

Maya buys a solid rubber cube to use as a protective support. Its volume is 125 cubic centimeters. If its edge length is x centimeters, then x³ = 125.

What is the cube's edge length? cm

12.

The students want a square business sign with an area of 2 square feet. Its side length x must satisfy x² = 2. Which description gives the exact side length?

A. The side is 1 foot, a rational length.B. The side is √2 feet, an irrational length.C. The side is 2 feet, a rational length.D. The side is √2 feet, a rational length.

13.

Select every expression whose value is irrational.

14.

On a practice sheet, Luis writes, “x³ = −125 has two solutions, −5 and 5, because taking a root always gives a positive and a negative answer.” Explain his mistake. Give the correct solution using a cube root symbol and check both proposed values.