4. Working with Exponential Expressions

Example 1: Simplify (2𝑥³𝑦²)² Solution: = 2² ⋅(𝑥³)² ⋅(𝑦²)² = 4𝑥⁶𝑦⁴

In this example, we want to simplify the expression (2x³y²)². To do this, we apply the Power of a Product Law, which states that when we raise a product to a power, we raise each factor in the product to that power. Here’s how it works step-by-step:

1. Start with (2x³y²)².
2. Applying the law, we calculate each part:
   • For the number 2, we compute 2², which equals 4.
   • For x³, we raise it to the power of 2, which gives us (x³)² = x⁶.
   • For y², we raise it to the power of 2, which results in (y²)² = y⁴.
3. Combine the results: 4x⁶y⁴. Therefore, (2x³y²)² simplifies to 4x⁶y⁴.

Think of it like preparing a recipe that requires 2x³y² for each serving. If you want to make it for 2 servings, you square all the ingredients in the recipe. So 2 becomes 2² (for the quantity), and x³ and y² get squared as well since you're multiplying each one by itself for the second serving.

Example 2: Simplify 54 ⋅53 52 Solution: = 54+3 = 57 = 57−2 = 55 = 3125

In this example, we simplify the expression 54 ⋅ 53 / 52. We use the Product of Powers Law, which tells us to add the exponents when multiplying. Here's the breakdown:

1. Start with 54 ⋅ 53. Using the product rule, we add the exponents: 4 + 3 = 7, leading to 57.
2. Now we have 57 / 52. For division, we apply the Quotient of Powers Law, where we subtract the exponents: 7 - 2 = 5.
3. Hence, our final result is 55, which equals 3125. Therefore, 54 ⋅ 53 / 52 simplifies to 3125.

Imagine you have 54 apples and then you get another 53 apples. When you put them together, you actually have 57 apples. Now, if you decide to share these 57 apples equally between 52 friends, the math tells you how many apples each friend gets. After the sharing, you'd still find that you have 3125 in terms of simpler 'apple bundles' left.

Example 3: Express with positive exponents only: 𝑥⁻²𝑦³ 𝑥⁴𝑦⁻¹ Solution: = 𝑦⁴ / 𝑥⁶

In this example, we want to convert the expression x⁻²y³ / x⁴y⁻¹ into one that contains only positive exponents. Here’s how to do this:

1. Start with the expression x⁻²y³ / x⁴y⁻¹.
2. Using the Negative Exponent Law, we recognize that x⁻² means 1/x² and y⁻¹ means 1/y.
3. The rule for division states we subtract the exponents of like bases. Thus, for x, we have -2 - 4 = -6, which gives us x⁻⁶. For y, we have 3 - (-1) = 3 + 1 = 4, resulting in y⁴.
4. Combining this, we can rewrite the expression as (y⁴ / x⁶). Thus, we've successfully expressed the entire fraction with positive exponents only.

Think of negative exponents like a debt. If you owe 2x (x⁻²), it means you are in the negative. But if you balance it by having 4x, it’s as if you are paying off that debt and rising to a total of y³. Once paid off, your remaining products only show as positive.

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