audio read-through Approximating Radicals

Need some basic understanding of radicals first? Practice with Radicals

Here, you will practice with radicals that don't come out ‘nicely’.

Examples

Find the two closest integers between which the given radical lies. Do not use the square root or cube root keys on a calculator.

Question: The number $\,\sqrt{7}\,$ lies between   ?   and   ?

Solution: We need a nonnegative number which, when squared, gives $\,7\,.$

$2^2 = 4\,$   ($\,2\,$ is too small)
$3^2 = 9\,$   ($\,3\,$ is too big)
Thus, $\,\sqrt{7}\,$ lies between $\,2\,$ and $\,3\,.$

Question: The number $\,\root 3\of{29}\,$ lies between   ?   and   ?

Solution: We need a number which, when cubed, gives $\,29\,.$

$3^3 = 27\,$   ($\,3\,$ is a bit too small)
$4^3 = 64\,$   ($\,4\,$ is too big)
Thus, $\,\root 3\of{29}\,$ lies between $\,3\,$ and $\,4\,.$

Question: The number $\,\root 3\of{-12}\,$ lies between   ?   and   ?

Solution: We need a number which, when cubed, gives $\,-12\,.$ The answer will be negative. Since it's easier to work with positive numbers, we first investigate $\,\root 3\of{12}\,$.

$2^3 = 8\,$   ($\,2\,$ is too small)
$3^3 = 27\,$   ($\,3\,$ is too big)
Thus, $\,\root 3\of{12}\,$ lies between $\,2\,$ and $\,3\,$, and $\,\root 3\of{-12}\,$ lies between $\,-3\,$ and $\,-2\,$.

For this web exercise, you must list the integers from least (farthest left) to greatest (farthest right).

Example
Question: Estimate $\,\sqrt{130}\,$ to the nearest tenth. Use a non-calculator approach.

Solution: To round to the tenths place, we must know if the digit in the hundredths place is $\,5\,$ or greater, or less than $\,5\,.$ As above, first determine that $\,\sqrt{130}\,$ is between $\,11\,$ and $\,12\,$. Then, use long multiplication:

$\,11.5^2 = 132.25\,$, so $\,11.5\,$ is a bit too big
$\,11.4^2 = 129.96\,$, so $\,11.4\,$ is a bit too small
Thus, $\,\sqrt{130}\,$ lies between $\,11.4\,$ and $\,11.5\,$.

Again using long multiplication:

$\,11.45^2 = 131.1025\,$, so $\,11.45\,$ is a bit too big.

Thus, the digit in the hundredths place must be less than $\,5\,$, and so the square root is closer to $\,11.4\,$.

Thus, $\,\sqrt{130}\approx 11.4\,$.

Practice

For this exercise, you must list the ‘lies between’ integers from least (farthest left) to greatest (farthest right).

lies between
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Concept Practice