Question

Write two different expressions that are equivalent to each of the expressions below. Be sure to show your work clearly and/or explain how you know that each of your expressions is equivalent.

a. (4x^5y^3z^2)^4
b. 1/x
c. (x+5)^2 - 7
d. 6(2x-7)

Answer 1

Two equivalent expressions for each given expression are provided, along with step-by-step explanations on how to obtain them.

Step-by-step explanation:

a. Two different equivalent expressions to (4x^5y^3z^2)^4 are:
- 4^4 * (x^5)^4 * (y^3)^4 * (z^2)^4
- (4 * x)^20 * (y^12) * (z^8)

To get these expressions, we raise each term inside the parentheses to the fourth power and apply the exponent rule that states (a * b)^n = a^n * b^n.
For example, (x^5)^4 = x^20, and (y^3)^4 = y^12.

b. Two different equivalent expressions to 1/x are:
- x^(-1)
- 1 * x^(-1)

These expressions use the negative exponent rule that states x^(-n) = 1/x^n.
So, 1/x can also be written as x^(-1).

c. Two different equivalent expressions to (x+5)^2 - 7 are:
- x^2 + 10x + 25 - 7
- x^2 + 10x + 18

We expand (x+5)^2 to x^2 + 10x + 25, and then subtract 7 to get x^2 + 10x + 18.

d. Two different equivalent expressions to 6(2x-7) are:
- 12x - 42
- -42 + 12x

These expressions use the distributive property of multiplication over addition or subtraction.