This answer key shows every step for the 25 problems on the square roots and cube roots worksheet (8.EE.A.2). Check your thinking against each solution, and remember that an equation \(x^{2}=p\) has two solutions while \(x^{3}=p\) has one.
Problem 1: Square root of 49
Ask which positive number times itself gives 49. Since \(7\cdot7=49\), the side of the square is 7 units.
\(\sqrt{49}=\)7
Problem 2: Square root of 144
Test nearby squares: \(11^{2}=121\) is too small and \(13^{2}=169\) is too big. Since \(12\cdot12=144\), \(\sqrt{144}=\)12.
Problem 3: Cube root of 27
Find a number used as a factor three times that makes 27. \(3\cdot3\cdot3=27\), so the edge is 3 units.
\(\sqrt[3]{27}=\)3
Problem 4: Cube root of 125
\(4^{3}=64\) is too small and \(6^{3}=216\) is too big. \(5\cdot5\cdot5=125\), so \(\sqrt[3]{125}=\)5.
Problem 5: Cube root of a negative
The cube of a negative number is negative. \((-4)(-4)=16\), and \(16\cdot(-4)=-64\). So \(\sqrt[3]{-64}=\)\(-4\).
Problem 6: Two square roots
Both \(9^{2}=81\) and \((-9)^{2}=81\). Take the square root of both sides: \(x=\sqrt{81}\) or \(x=-\sqrt{81}\).
\(x=9\) or \(x=-9\)
Problem 7: A cube equation
Take the cube root of both sides: \(x=\sqrt[3]{216}\). Since \(6\cdot6\cdot6=216\), \(x=6\). There is only one solution.
Problem 8: Another square equation
\(13\cdot13=169\), so \(x=13\) or \(x=-13\). Check: \((-13)^{2}=169\).
\(x=13\) or \(x=-13\)
Problem 9: A negative cube
\(7^{3}=343\), so \((-7)^{3}=-343\). A cube equation has one solution: \(x=-7\).
Problem 10: Fraction square
Take the square root of the numerator and the denominator: \(\sqrt{4}=2\) and \(\sqrt{25}=5\). Check: \(\frac{2}{5}\cdot\frac{2}{5}=\frac{4}{25}\).
\(x=\frac{2}{5}\) or \(x=-\frac{2}{5}\)
Problem 11: Fraction cube
\(\sqrt[3]{8}=2\) and \(\sqrt[3]{27}=3\). Check: \(\frac{2}{3}\cdot\frac{2}{3}\cdot\frac{2}{3}=\frac{8}{27}\).
\(x=\frac{2}{3}\)
Problem 12: Decimal square
Since \(0.64=\frac{64}{100}\), the root is \(\frac{8}{10}=0.8\). Check: \(0.8\cdot0.8=0.64\).
\(x=0.8\) or \(x=-0.8\)
Problem 13: Decimal cube
\(0.001=\frac{1}{1000}\) and \(\sqrt[3]{1000}=10\), so \(x=\frac{1}{10}\). Check: \(0.1\cdot0.1\cdot0.1=0.001\).
\(x=0.1\)
Problem 14: Undo, then take the root
Subtract 6 from both sides: \(x^{2}=64\). Then take square roots: \(x=8\) or \(x=-8\). Check: \(8^{2}+6=70\).
\(x=8\) or \(x=-8\)
Problem 15: Between two integers
\(7^{2}=49\) and \(8^{2}=64\). Since \(49<50<64\), \(\sqrt{49}<\sqrt{50}<\sqrt{64}\).
\(\sqrt{50}\) is between 7 and 8, a little closer to 7.
Problem 16: Cube root estimate
\(4^{3}=64\) and \(5^{3}=125\). Since \(64<100<125\), the cube root is between them.
Between 4 and 5
Problem 17: Which number is irrational?
\(\sqrt{36}=6\), \(\sqrt[3]{8}=2\) and \(\sqrt{100}=10\) are whole numbers, so they are rational. The number \(\sqrt{2}\) is not a whole number, and no fraction squares to exactly 2. Its decimal, 1.41421356 and more digits, never ends or repeats.
\(\sqrt{2}\) is irrational.
Problem 18: Compare a root and a decimal
Square 5.5: \(5.5^{2}=30.25\). Because 30 is less than 30.25, the number whose square is 30 is less than 5.5.
\(\sqrt{30}<5.5\)
Problem 19: Square garden fence
The side is \(\sqrt{196}\). Since \(14\cdot14=196\), each side is 14 ft.
Fence: \(4\cdot14=56\) ft.
Side 14 ft; fence 56 ft
Problem 20: Gift box
The edge is \(\sqrt[3]{343}=7\) in because \(7^{3}=343\). One face is a 7 in by 7 in square: \(7^{2}=49\) square inches.
Edge 7 in; face area 49 sq in
Problem 21: Square rug fringe
Side: \(\sqrt{20.25}=4.5\), since \(4.5\cdot4.5=20.25\).
Perimeter: \(4\cdot4.5=18\) ft. Cost: \(18\cdot1.50=27\).
Side 4.5 ft; fringe costs $27.00
Problem 22: Storage cube
The edge is \(\sqrt[3]{1728}\). Since \(12^{3}=12\cdot144=1728\), the edge is 12 inches. There are 12 inches in a foot, so the edge is 1 foot.
12 in, which is 1 ft
Problem 23: Playground laps
Side: \(\sqrt{2500}=50\) ft, since \(50\cdot50=2500\). One lap: \(4\cdot50=200\) ft. Three laps: \(3\cdot200=600\) ft.
600 ft
Problem 24: Side of a square patio
The side is \(\sqrt{50}\), between 7 and 8. Try 7.0: \(7.0^{2}=49\). Try 7.1: \(7.1^{2}=50.41\). Since 50 is 1.00 above 49 but only 0.41 below 50.41, 7.1 is closer.
About 7.1 ft
Problem 25: Edge of a small cube
The edge is \(\sqrt[3]{40}\), between 3 and 4. Try 3.4: \(3.4^{3}=39.304\). Try 3.5: \(3.5^{3}=42.875\). The number 40 is only 0.696 above 39.304, but 2.875 below 42.875.
About 3.4 in
See every chapter on the Grade 8 math page.