Chapter 6.1, Problem 57E

a.

To determine

To find: The properties of exponents used to simplify an expression for the volume of each cube.

Answer to Problem 57E

We use rule of power of a product property of exponents. ${(ab)}^x=a^x*b^x$

Explanation of Solution

Given information: There are two cubes with side lengths 2x and 6x .

Formula used: Volume of cube is $a^3$ . Where ‘ a’ is the side length of each cube.

Calculation:

Using formula to find volume of a cube,

Volume of small cube= ${(2x)}^3$

Let’s use power of a product property of exponents

  $=2^3{x}^3$

  $=8x^3$

Volume of big cube $={(6x)}^3$

  $=6^3{x}^3$

  $=216x^3$

b.

To determine

To Explain: how to use power of quotient property to explain how many times the greater cube has volume than small cube.

Answer to Problem 57E

We do subtract the exponents to simplify the exponents which gives Big cube is 27 times more volume than small cube.

Explanation of Solution

Given information: The volume of small cube is $8x^3$ and volume of big cube is $216x^3$

Calculation:

Volume of big cube= $216x^3$

Volume of small cube= $8x^3$

Now, let’s find how many times big cube is greater than small cube.

  $=\frac{216x^3}{8x^3}$

  $=\frac{216x^{3-3}}{8}$

  $=27x^0$

   =27 (here $x^0$ is 1)

We do subtract the exponents to simplify which gives Big cube is 27 times more volume than small cube.