Question

to estimate the cube root of a non - perfect cube.
cubes
estimate the cube roots by finding the perfect cubes that are less and greater than the number. find the cube roots of the perfect cubes for your estimate. then, circle the integer that the estimate is closest to. the first one is done for you.

	consecutive cubes	in between the integers
$sqrt3{30}$
$sqrt3{123}$
$sqrt3{63}$
$sqrt3{500}$
$sqrt3{237}$
$sqrt3{195}$
$sqrt3{892}$
$sqrt3{1400}$
$sqrt3{23}$
$sqrt3{317}$

Explanation:

Step1: Recall perfect - cube values

The perfect - cube values are \(1^3 = 1\), \(2^3=8\), \(3^3 = 27\), \(4^3=64\), \(5^3 = 125\), \(6^3=216\), \(7^3 = 343\), \(8^3=512\), \(9^3 = 729\), \(10^3=1000\), \(11^3 = 1331\), \(12^3=1728\).

Step2: Estimate \(\sqrt[3]{30}\)

Since \(3^3=27\) and \(4^3 = 64\), and \(27<30<64\), \(\sqrt[3]{30}\) is between \(3\) and \(4\), and closer to \(3\).

Step3: Estimate \(\sqrt[3]{123}\)

Since \(4^3 = 64\) and \(5^3=125\), and \(64<123<125\), \(\sqrt[3]{123}\) is between \(4\) and \(5\), and closer to \(5\).

Step4: Estimate \(\sqrt[3]{63}\)

Since \(4^3 = 64\) and \(3^3=27\), and \(27<63<64\), \(\sqrt[3]{63}\) is between \(3\) and \(4\), and closer to \(4\).

Step5: Estimate \(\sqrt[3]{500}\)

Since \(7^3 = 343\) and \(8^3=512\), and \(343<500<512\), \(\sqrt[3]{500}\) is between \(7\) and \(8\), and closer to \(8\).

Step6: Estimate \(\sqrt[3]{237}\)

Since \(6^3 = 216\) and \(7^3=343\), and \(216<237<343\), \(\sqrt[3]{237}\) is between \(6\) and \(7\), and closer to \(6\).

Step7: Estimate \(\sqrt[3]{195}\)

Since \(5^3 = 125\) and \(6^3=216\), and \(125<195<216\), \(\sqrt[3]{195}\) is between \(5\) and \(6\), and closer to \(6\).

Step8: Estimate \(\sqrt[3]{892}\)

Since \(9^3 = 729\) and \(10^3=1000\), and \(729<892<1000\), \(\sqrt[3]{892}\) is between \(9\) and \(10\), and closer to \(9\).

Step9: Estimate \(\sqrt[3]{1400}\)

Since \(11^3 = 1331\) and \(12^3=1728\), and \(1331<1400<1728\), \(\sqrt[3]{1400}\) is between \(11\) and \(12\), and closer to \(11\).

Step10: Estimate \(\sqrt[3]{23}\)

Since \(2^3 = 8\) and \(3^3=27\), and \(8<23<27\), \(\sqrt[3]{23}\) is between \(2\) and \(3\), and closer to \(3\).

Step11: Estimate \(\sqrt[3]{317}\)

Since \(6^3 = 216\) and \(7^3=343\), and \(216<317<343\), \(\sqrt[3]{317}\) is between \(6\) and \(7\), and closer to \(7\).

Answer:

\(\sqrt[3]{}\)	Consecutive Cubes	In Between the Integers
\(\sqrt[3]{123}\)	\(64\) and \(125\)	\(4\) and \(5\) (circle \(5\))
\(\sqrt[3]{63}\)	\(27\) and \(64\)	\(3\) and \(4\) (circle \(4\))
\(\sqrt[3]{500}\)	\(343\) and \(512\)	\(7\) and \(8\) (circle \(8\))
\(\sqrt[3]{237}\)	\(216\) and \(343\)	\(6\) and \(7\) (circle \(6\))
\(\sqrt[3]{195}\)	\(125\) and \(216\)	\(5\) and \(6\) (circle \(6\))
\(\sqrt[3]{892}\)	\(729\) and \(1000\)	\(9\) and \(10\) (circle \(9\))
\(\sqrt[3]{1400}\)	\(1331\) and \(1728\)	\(11\) and \(12\) (circle \(11\))
\(\sqrt[3]{23}\)	\(8\) and \(27\)	\(2\) and \(3\) (circle \(3\))
\(\sqrt[3]{317}\)	\(216\) and \(343\)	\(6\) and \(7\) (circle \(7\))