Square Roots: Additional Practice (Set 2)

A radical acts like a grouped quantity, so evaluate the root first, then follow the order of operations (multiply before add).

$\sqrt{25} = 5, \qquad 3 + 5 \times 2 = 3 + 10 = 13$

The multiplication comes before the addition.

Find the perfect squares bracketing $200$. The square $196 = 14^2$ is below, and $225 = 15^2$ is above.

$196 < 200 < 225 \quad\Rightarrow\quad 14 < \sqrt{200} < 15$

So $\sqrt{200}$ lies between $14$ and $15$.

Evaluate each perfect-square root, then subtract.

$\sqrt{144} - \sqrt{81} = 12 - 9 = 3$

This is not $\sqrt{144 - 81} = \sqrt{63}$; roots do not combine across subtraction.

Work out what is inside the radical first, then take the root of the single number.

$\sqrt{100 - 36} = \sqrt{64} = 8$

This is not $\sqrt{100} - \sqrt{36} = 10 - 6 = 4$; a root does not split across subtraction.

First simplify $\sqrt{18}$, then multiply by the $2$ in front. Since $18 = 9 \times 2$,

$\sqrt{18} = \sqrt{9}\,\sqrt{2} = 3\sqrt{2}, \qquad 2\sqrt{18} = 2 \times 3\sqrt{2} = 6\sqrt{2}$

The coefficient $2$ multiplies the $3$ to give $6\sqrt{2}$.

Squaring undoes the root, so $n$ is the square of $20$.

$n = 20^2 = 400$

Check: $\sqrt{400} = 20$.

The area of a square is the side times itself, so the side is the square root of the area.

$\text{side} = \sqrt{169} = 13 \text{ m}$

The side is $13$ meters, since $13^2 = 169$. It is not half the area.

Rooting undoes squaring, so $\sqrt{a^2} = a$ for a non-negative number $a$.

$\sqrt{15^2} = 15$

No need to compute $15^2 = 225$ first; the square and the root cancel.

Find the non-negative number whose square is $1.44$. Try $1.2$.

$1.2^2 = 1.2 \times 1.2 = 1.44 \quad\Rightarrow\quad \sqrt{1.44} = 1.2$

A decimal radicand still has a square root: the number that squares to it.

Evaluate the root first, then follow the order of operations (multiply before add).

$\sqrt{49} = 7, \qquad 2 \times 7 + 1 = 14 + 1 = 15$

The multiplication comes before the addition.

Both terms share the radical part $\sqrt{5}$, so they add like quantities of the same thing.

$4\sqrt{5} + 2\sqrt{5} = (4 + 2)\sqrt{5} = 6\sqrt{5}$

The $\sqrt{5}$ stays as the common factor; only the coefficients add.

The trial squares bracket $55$, so $7.4 < \sqrt{55} < 7.5$. Compare distances.

$55 - 54.76 = 0.24, \qquad 56.25 - 55 = 1.25$

The radicand is much closer to $54.76$, so $\sqrt{55} \approx 7.4$ to the nearest tenth.