We know how to find the square root of a perfect square. But how do we find square roots of numbers that are not perfect squares?
For example, what is the square root of
Even though is not a perfect square, is still a number. Such numbers are called surds. Surds are a new type of number that cannot be written as a whole number or regular fraction!
Surds can, however, be written as infinite, non-repeating decimals.
We can find a decimal approximation to the value of using the square root button on our calculator.
Different calculator models work differently. However, one of the following sequences should work with your calculator:
Press type , and then press .
Alternatively, you may need to type first, then press followed by .
If you find yourself stuck, search for an online video that shows how to use the square root button with your calculator model.
When we do this, we get the following on our calculator's display.
The value given by the calculator is an approximation to the true value of In reality, the digits continue forever, and the pattern of decimal digits does not repeat. It is impossible to express precisely using decimals.
What is the value of rounded to decimal places?
Using a calculator, we find that Now, we round to decimal places:
What is $\sqrt{33}$ rounded to $1$ decimal place?
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What is $\sqrt{8}$ rounded to $2$ decimal places?
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A number is not a surd if we can write it as a positive or negative whole number or fraction.
For example, is not a surd because it is a perfect square and can be expressed as a whole number:
Similarly, is not a surd because it can be written as a regular fraction (i.e., as the ratio of two whole numbers):
Finally, we cannot have surds involving square roots of negative numbers. In other words, the numbers
are not surds because they are not real numbers!
Which of the following numbers are surds?
A surd is a number containing a radical symbol that cannot be written as a positive or negative whole number or fraction.
With that in mind, let's look at each number in turn.
is a surd. Using a calculator, we find that
is a surd. Using a calculator, we find that
is NOT a surd. Since is a perfect square, we can get rid of the square root symbol () and obtain a whole number:
Therefore, the correct answer is: and only.
Which of the following numbers are surds?
\[ \sqrt{121},\qquad \sqrt{225},\qquad \sqrt{19} \]
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a
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$\sqrt{121}$ and $\sqrt{19}$ only |
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b
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$\sqrt{121},$ $\sqrt{225},$ and $\sqrt{19}$ |
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c
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$\sqrt{121}$ only |
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d
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$\sqrt{225}$ only |
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e
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$\sqrt{19}$ only |
Which of the following numbers are NOT surds?
\[ \sqrt{4},\qquad \sqrt{14},\qquad \sqrt{19} \]
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a
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$\sqrt{14}$ and $\sqrt{19}$ only |
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b
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$\sqrt{4}$ and $\sqrt{14}$ only |
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c
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$\sqrt{14}$ only |
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d
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$\sqrt{19}$ only |
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e
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$\sqrt{4}$ only |
Which of the following numbers are surds?
A surd is a number containing a radical symbol that cannot be written as a positive or negative whole number or fraction.
With that in mind, let's look at each number in turn.
is a surd. Using a calculator, we find that
is NOT a surd. Since and are perfect squares, we can get rid of the square root symbol and obtain a regular fraction.
is a surd. Using a calculator, we find that
Therefore, the correct answer is: and only.
Which of the following numbers are surds?
\[ \sqrt{\dfrac{1}{2}}, \qquad \sqrt{\dfrac{1}{3}}, \qquad \sqrt{\dfrac{1}{4}} \]
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a
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$\sqrt{\dfrac{1}{2}}$ and $\sqrt{\dfrac{1}{3}}$ only |
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b
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All the numbers are surds. |
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c
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$\sqrt{\dfrac{1}{3}}$ only |
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d
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$\sqrt{\dfrac{1}{2}}$ and $\sqrt{\dfrac{1}{4}}$ only |
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e
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$\sqrt{\dfrac{1}{2}}$ only |
Which of the following numbers are NOT surds? \[ \sqrt{\dfrac{9}{7}}, \qquad \sqrt{\dfrac{3}{10}}, \qquad \sqrt{\dfrac{25}{4}} \]
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a
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$\sqrt{\dfrac{9}{7}}$ only |
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b
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$\sqrt{\dfrac{3}{10}}$ and $\sqrt{\dfrac{25}{4}}$ only |
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c
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$\sqrt{\dfrac{25}{4}}$ only |
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d
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$\sqrt{\dfrac{3}{10}}$ only |
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e
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$\sqrt{\dfrac{9}{7}}$ and $\sqrt{\dfrac{3}{10}}$ only |
Similarly, we can have surds involving the cube root sign (). For example, But is not a surd since we can get rid of the cube root:
Most calculators have a "cube root button" that can be used to compute cube roots.
Note that we can have surds involving cube roots () of negative numbers, such as because it's fine to take the cube root of a negative number.
Which of the following numbers are surds?
A surd is a number containing a radical symbol that cannot be written as a positive or negative whole number or fraction.
Surds can be square roots , cube roots or higher roots such as etc.
With that in mind, let's look at each number in turn.
is a surd. Using a calculator, we find that
is NOT a surd. Since and are perfect cubes, we can get rid of the cube root symbol and obtain a regular fraction.
is a surd. Using a calculator, we find that
Therefore, the correct answer is: and only.
Which of the following numbers are surds?
\[ \sqrt[3]{7},\qquad \sqrt[3]{8},\qquad \sqrt[3]{9} \]
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a
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$\sqrt[3]{7}$ only |
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b
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$\sqrt[3]{7}$ and $\sqrt[3]{9}$ only |
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c
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$\sqrt[3]{8}$ only |
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d
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$\sqrt[3]{9}$ only |
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e
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$\sqrt[3]{8}$ and $\sqrt[3]{9}$ only |
Which of the following numbers are surds?
\[ \sqrt[\ 3\ ] {\dfrac{1}{5}}, \qquad \sqrt[3\ ]{\dfrac{1}{64}}, \qquad \sqrt[3\ ]{\dfrac{8}{27}} \]
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a
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$\sqrt[3\ ]{\dfrac{1}{5}}$ only |
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b
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$\sqrt[3\ ]{\dfrac{1}{5}}$ and $\sqrt[3\ ]{\dfrac{8}{27}}$ only |
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c
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$\sqrt[3\ ]{\dfrac{8}{27}}$ only |
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d
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$\sqrt[3\ ]{\dfrac{1}{64}}$ only |
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e
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$\sqrt[3\ ]{\dfrac{1}{64}}$ and $\sqrt[3]{\dfrac{8}{27}}$ only |