Question

squares
squaring numbers, polynomials and rational expressions is something you have done many times in algebra. we use the word “squaring” to mean multiplying a number or expression by itself, because this is what we do to find the areas of squares.
find the area of each shaded square.
write an expression for the area of each square.
to find the area of any square all we need to know is the length of one side. it is also possible to find the length of a side if we know the area. try to figure out the lengths of the sides of each square below.
we call the area the square of the length of a side. we say that the length of a side is the square root of the area. in this book we will study square roots. knowing about square roots enables us to solve some quadratic equations we could not solve by factoring.

Explanation:

Step1: Recall area formula for square

The area formula of a square is $A = s^2$, where $s$ is the side - length of the square.

Step2: Calculate areas for given side - lengths

For side - length $s = 5$:

$A=5^2 = 25$

For side - length $s = 43$:

$A = 43^2=1849$

For side - length $s = 1.2$:

$A=1.2^2 = 1.44$

For side - length $s=\frac{1}{3}$:

$A = (\frac{1}{3})^2=\frac{1}{9}$

For side - length $s=\frac{2}{3}$:

$A=(\frac{2}{3})^2=\frac{4}{9}$

For side - length $s=x + 3$:

$A=(x + 3)^2=x^{2}+6x + 9$

For side - length $s = m$:

$A=m^2$

For side - length $s=2x-1$:

$A=(2x - 1)^2=4x^{2}-4x + 1$

For side - length $s=\frac{6y}{z}$:

$A = (\frac{6y}{z})^2=\frac{36y^{2}}{z^{2}}$

Step3: Calculate side - lengths for given areas

For $A = 9$:

Since $s=\sqrt{A}$, then $s=\sqrt{9}=3$

For $A=\frac{1}{4}$:

$s=\sqrt{\frac{1}{4}}=\frac{1}{2}$

For $A = 0.09$:

$s=\sqrt{0.09}=0.3$

For $A = 576$:

$s=\sqrt{576}=24$

For $A=\frac{16}{25}$:

$s=\sqrt{\frac{16}{25}}=\frac{4}{5}$

For $A = 144$:

$s=\sqrt{144}=12$

Answer:

Areas for first set of squares: 25, 1849, 1.44, $\frac{1}{9}$, $\frac{4}{9}$
Expressions for areas of second set of squares: $x^{2}+6x + 9$, $m^{2}$, $4x^{2}-4x + 1$, $\frac{36y^{2}}{z^{2}}$
Side - lengths for third set of squares: 3, $\frac{1}{2}$, 0.3, 24, $\frac{4}{5}$, 12