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
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$\,\sqrt{7}\,$ lies between ? and ?
Solution:
We need a nonnegative number which, when squared, gives $\,7\,$.
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$2^2 = 4\,$ ($\,2\,$ is too small)
$3^2 = 9\,$ ($\,3\,$ is too big)
Thus,
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$\,\sqrt{7}\,$ lies between $\,2\,$ and $\,3\,$.
Question:
The number
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$\,\root 3\of{29}\,$ lies between ? and ?
Solution:
We need a number which, when cubed, gives
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$\,29\,$.
$3^3 = 27\,$ ($\,3\,$ is a bit too small)
$4^3 = 64\,$ ($\,4\,$ is too big)
Thus,
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$\,\root 3\of{29}\,$ lies between $\,3\,$ and $\,4\,$.
Question:
The number
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$\,\root 3\of{-12}\,$ lies between ? and ?
Solution:
We need a number which, when cubed, gives
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$\,-12\,$.
The answer will be negative.
Since it's easier to work with positive numbers, we first investigate
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$\,\root 3\of{12}\,$.
$2^3 = 8\,$ ($\,2\,$ is too small)
$3^3 = 27\,$ ($\,3\,$ is too big)
Thus,
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$\,\root 3\of{12}\,$ lies between $\,2\,$ and $\,3\,$,
and
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$\,\root 3\of{-12}\,$ lies between $\,-3\,$ and $\,-2\,$.
For this web exercise, you MUST must list the integers from least (farthest left)
to greatest (farthest right).
EXAMPLE:
Question:
Estimate
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$\,\sqrt{130}\,$ 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
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$\,\sqrt{130}\,$
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,
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$\,\sqrt{130}\,$ 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,
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$\,\sqrt{130}\approx 11.4\,$.
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Renaming Square Roots