(6x^4y^5)^2
Understand the Problem
The question is asking to simplify the expression (6x^4y^5)^2. This involves applying the power of a product rule in exponents, which states that when raising a product to a power, you raise each factor to the power separately.
Answer
The simplified expression is \( 36x^8y^{10} \).
Answer for screen readers
The simplified expression is ( 36x^8y^{10} ).
Steps to Solve
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Apply the Power of a Product Rule
To simplify $(6x^4y^5)^2$, we need to apply the power of a product rule. This rule states that when raising a product to a power, each factor must be raised to that power. -
Break Down the Expression
Using the rule:
$$(6x^4y^5)^2 = 6^2 \cdot (x^4)^2 \cdot (y^5)^2$$
Now we will simplify each term separately. -
Simplify Each Term
- Simplifying $6^2$ gives us $36$.
- Simplifying $(x^4)^2$ yields $x^{4 \cdot 2} = x^8$.
- Simplifying $(y^5)^2$ results in $y^{5 \cdot 2} = y^{10}$.
Thus, we have: $$36 \cdot x^8 \cdot y^{10}$$
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Combine the Terms
Putting it all together, we get the simplified expression:
$$36x^8y^{10}$$
The simplified expression is ( 36x^8y^{10} ).
More Information
This expression shows that we multiplied the constants and applied the exponent rules to the variables correctly. The power of a product rule is fundamental in algebra, often used in polynomial expressions.
Tips
- Forgetting to Square Each Factor: One common mistake is forgetting to apply the exponent to each variable and number. Always remember to apply the exponent to every component in the expression.
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