Square roots
Lucía's bedroom is square and has a floor area of $16$ m². How long is each wall? You need the opposite operation to squaring: the square root.
What a square root is
Change the area and see how the side of the square changes.
Zero is also a square number: $\sqrt{0} = 0$. Negative numbers have no real square root.
Work out mentally the value of the following exact square roots using multiplication tables: a) $\sqrt{36}$ b) $\sqrt{64}$ c) $\sqrt{81}$ d) $\sqrt{100}$ e) $\sqrt{121}$
Work out mentally the value of the following exact square roots using multiplication tables:
- a) $\sqrt{36}$
- b) $\sqrt{64}$
- c) $\sqrt{81}$
- d) $\sqrt{100}$
- e) $\sqrt{121}$
Match each power with its corresponding root in your notebook, as in the example: Example: $7^2 = 49 \rightarrow \sqrt{49} = 7$* a) $4^2 = 16$ b) $9^2 = 81$ c)…
Match each power with its corresponding root in your notebook, as in the example:
Example: $7^2 = 49 \rightarrow \sqrt{49} = 7$
- a) $4^2 = 16$
- b) $9^2 = 81$
- c) $12^2 = 144$
- d) $15^2 = 225$
When the root is not exact
Not every number is a perfect square. For those numbers we look for the integer square root: the largest whole number whose square does not exceed the radicand.
State between which two consecutive natural numbers the following square roots lie. Look at the example: Example: $\sqrt{10}$ lies between $3$ and $4$ because $3^2=9$…
State between which two consecutive natural numbers the following square roots lie. Look at the example:
Example: $\sqrt{10}$ lies between $3$ and $4$ because $3^2=9$ and $4^2=16$.
- a) $\sqrt{20}$
- b) $\sqrt{45}$
- c) $\sqrt{70}$
- d) $\sqrt{95}$
Knowing that a natural number $N$ has an integer square root $a = 45$, calculate the largest value that this number $N$ can take and what its remainder is in that case.…
Knowing that a natural number $N$ has an integer square root $a = 45$, calculate the largest value that this number $N$ can take and what its remainder is in that case. Take into account the property that relates the remainder to the integer root so that the approximation is maximal. In the figure (not exactly to scale in the gaps) you have the square of side $a$ and the next square, of side $a+1$.
Properties of roots
Apply the root of a product rule to calculate the following square roots more easily, as shown in the example: Example:…
Apply the root of a product rule to calculate the following square roots more easily, as shown in the example:
Example: $\sqrt{4 \cdot 25} = \sqrt{4} \cdot \sqrt{25} = 2 \cdot 5 = 10$
- a) $\sqrt{9 \cdot 100}$
- b) $\sqrt{16 \cdot 25}$
- c) $\sqrt{4 \cdot 81}$
Work out the value of the following expressions, respecting the order of operations: a) $3 \cdot \sqrt{81} - 2^2 \cdot (10 - \sqrt{49})$ b)…
Work out the value of the following expressions, respecting the order of operations:
- a) $3 \cdot \sqrt{81} - 2^2 \cdot (10 - \sqrt{49})$
- b) $(\sqrt{100} - 2^3)^2 + \sqrt{144} : 3$
Combined operations with roots
The square root is calculated at the same level as powers in the order of operations.
Solve the following combined operations, respecting the order of operations: a) $10 + \sqrt{49}$ b) $3 \cdot \sqrt{25} - 2^2$ c) $\sqrt{64} : 2 + 5$ d)…
Solve the following combined operations, respecting the order of operations:
- a) $10 + \sqrt{49}$
- b) $3 \cdot \sqrt{25} - 2^2$
- c) $\sqrt{64} : 2 + 5$
- d) $(\sqrt{100} - 4) \cdot 3$
A logistics company wants to store $196$ packages of medical supplies in a warehouse with a square floor plan. If the packages must be placed forming a perfect square on…
A logistics company wants to store $196$ packages of medical supplies in a warehouse with a square floor plan. If the packages must be placed forming a perfect square on the floor, how many packages will be on each side? If they later receive another $30$ packages and want to form the largest possible square again, how many packages will be left over?
Work out the value of the following combined expression, paying attention to the order of operations and the properties of roots: $$ \sqrt{144} : (-3) + [(-2)^3 \cdot…
Work out the value of the following combined expression, paying attention to the order of operations and the properties of roots:
$$ \sqrt{144} : (-3) + [(-2)^3 \cdot \sqrt{25} - (-4)^2] - \sqrt{169 \cdot 4} $$Back to the beginning: Each wall of Lucía's bedroom is $\sqrt{16} = 4$ m long.