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Understanding (4x³y⁵)²
In mathematics, the expression (4x³y⁵)² represents the square of the entire term 4x³y⁵. Let's break down the steps to understand how to simplify this expression:
Understanding the Concept of Squaring
Squaring a term means multiplying it by itself. In this case, we are squaring the entire expression (4x³y⁵), which means we are multiplying it by itself:
(4x³y⁵)² = (4x³y⁵) * (4x³y⁵)
Applying the Power of a Product Rule
The power of a product rule states that when you raise a product to a power, you raise each factor to that power. In this case, we have a product of four factors: 4, x³, y⁵, and another 4, x³, and y⁵.
So, we apply the rule to each factor individually:
(4x³y⁵)² = 4² * (x³)² * (y⁵)²
Simplifying the Expression
Now, we simplify each factor by applying the power of a power rule. This rule states that when raising a power to another power, you multiply the exponents.
- 4² = 16
- (x³)² = x⁶ (3 * 2 = 6)
- (y⁵)² = y¹⁰ (5 * 2 = 10)
Therefore, the simplified expression is:
(4x³y⁵)² = 16x⁶y¹⁰
Conclusion
By understanding the fundamental concepts of squaring and applying the power of a product and power of a power rules, we can effectively simplify expressions like (4x³y⁵)². This understanding is crucial for solving more complex mathematical problems involving exponents.