Powers: Using additional geometric shapes

Using the formula for the area of a square whose side is b:

$S=b^2$In the picture, we are presented with three squares whose sides from left to right have a length of 6, 3, and 4 respectively:

Therefore the areas are:

$S_1=3^2,\hspace{4pt}S_2=6^2,\hspace{4pt}S_3=4^2$square units respectively,

Consequently the total area of the shape, composed of the three squares, is as follows:

$S_{\text{total}}=S_1+S_2+S_3=3^2+6^2+4^2$square units

To conclude, we recognise through the rules of substitution and addition that the correct answer is answer C.

Let's denote the initial cube's edge length as x,

The formula for the volume of a cube with edge length b is:

$V=b^3$

therefore the volume of the initial cube (meaning before increasing its edge) is:

$V_1=x^3$

Now we'll increase the cube's edge by a factor of 6, meaning the edge length is now: 6x, therefore the volume of the new cube is:

$V_2=(6x)^3=6^3x^3$

where in the second step we simplified the expression for the new cube's volume using the power rule for multiplication in parentheses:

$(z\cdot y)^n=z^n\cdot y^n$

and we applied the power to each term in the parentheses multiplication,

Next we'll answer the question that was asked - "By what factor did the cube's volume increase", meaning - by what factor do we multiply the old cube's volume (before increasing its edge) to get the new cube's volume?

Therefore to answer this question we simply divide the new cube's volume by the old cube's volume:

$\frac{V_2}{V_1}=\frac{6^3x^3}{x^3}=6^3$

where in the first step we substituted the expressions for the volumes of the old and new cubes that we got above, and in the second step we reduced the common factor between the numerator and denominator,

Therefore we got that the cube's volume increased by a factor of -$6^3$when we increased its edge by a factor of 6,

therefore the correct answer is b.

We will refer to two separate areas: the area of the square with side y and the total area of the four squares with sides x,

We'll use the formula for the area of a square with side b:

$S=b^2$and therefore when applying it to the problem, we get that the area of the square with side y in the drawing is:

$S_1=y^2$Next, we'll calculate the area of the square with side x in the drawing:

$S_2=x^2$and to get the total area of the four squares in the drawing, we'll multiply this area by 4:

$4S_2=4x^2$Therefore, the area of the required figure in the problem, which includes the area of the square with side y and the area of the four squares with side x is:

$S_1+4S_2=y^2+4x^2$Therefore, the correct answer is A.