**Introduction**

Equivalent Expressions problems are common on the digital SAT, so it’s important to have a strategy for how to approach them. These problems ask you to pick an answer choice that is equivalent to a given expression. To solve these questions, you might need to add, subtract, multiply, divide, foil, factor, or simplify exponents and logarithms. However, there’s a helpful trick using Desmos that can decrease the number of overall steps, reduce the complexity, and help us avoid making mistakes!

**Equivalent Expressions Problems:** Ask you to pick an answer choice that is equivalent to a given expression.

**Expression:** any mathematical statement with numbers, variables, and arithmetic symbols in between (no equals signs – that would be an equation)

The goal with these questions is to pick the matching expression. This may involve a lot of arithmetic. However, there is a Desmos Trick to make it easier.

**Desmos Trick:**

- Type the original expression into Desmos
- If it gives you a graph, type in the answer choices until one of them gives you the same graph
- If it doesn’t, set it equal to an arbitrary number, which will now give a graph. Then, type in the answer choices and set them equal to
**the same number**(if feeling advanced, set it equal to**the same constant**) - If the expression doesn’t have x and y, make up numbers!

**Example One (just x): **

**Equivalent Expressions**

Which of the following expressions is equivalent to (81 *x^*4)^1/4?

a. |9x|

b.|3x|

c. 3√(3x)

d. |18x|

Answer choice B is a match because it overlaps the original expression (black). But, we will check the rest anyway:

I turned off answer choice B so that we could see the original expression (black). Answer choices C and D are not matches, so B is the correct answer.

**Example Two (x and y):**

**Equivalent Expressions**

Which of the following expressions is equivalent to (64 *x^*3 *y^*5)^1/3 , where x>0 and y>0?

a. 8 *x**y^(*5/3)

b. 4 *x**y^(*5/3)

c. 12 *x^*3 *y^*5

d. 16 *x^(*3/4) *y^*5

Step 1: Graph the original expression. Unfortunately, this one doesn’t give us a graph. So, I’m going to set it equal to an arbitrary number. I’ll use 2. When I enter the number, I now get a graph. Now, I can start checking the answer choices by setting them equal to the same number.

Step 2: Check answer choices. When I set answer choice A equal to 2, I see that it is close but not a perfect match.

Answer choice B looks like a match! Let’s check the rest.

C and D also don’t look like matches, so the correct answer is B!

**Example Three (not x and y):**

Step 1: Let’s start by typing out the original expression into Desmos and making up some numbers. We’ll set each variable to a different number. This tells us that A = 3.

Step 2: Now, we’ll type in the answer choices. Since each answer choice says “H = “, we can just type in the second half. When we type out answer choice A, it’s equal to -4. Since H=4, this isn’t our answer. We’re looking for the answer that equals H, which equals 4.

Step 3: Type in the rest of the answers and see which one matches. As we can see, answer choice C is the only one that equals 4. So, C is the correct answer.

**Now it’s your turn! **

Here are some practice problems for you to try out!

**Conclusion**

Practicing these problems using the Desmos trick will help you become more comfortable with identifying equivalent expressions quickly and accurately. Keep practicing, and you’ll master this skill in no time!

**Answers:**

- B
- B
- B
- C
- D
- A
- A
- C
- C
- B
- A
- A
- A