(y^3 * 2^5)^2 = ?
Understand the Problem
The question is asking for the equivalent expression of ( y^3 imes 2^5 )^2 using the properties of exponents. The goal is to simplify the expression correctly.
Answer
$y^6 \cdot 2^{10}$
Answer for screen readers
The equivalent expression is $y^6 \cdot 2^{10}$.
Steps to Solve
- Apply the Power of a Product Rule
Using the property of exponents that states $(a \cdot b)^n = a^n \cdot b^n$, we can express the given equation as:
$$ (y^3 \cdot 2^5)^2 = (y^3)^2 \cdot (2^5)^2 $$
- Use the Power Rule on Each Term
Now, we apply the power rule, which states that $(x^m)^n = x^{m \cdot n}$. Therefore:
$$ (y^3)^2 = y^{3 \cdot 2} = y^6 $$
and
$$ (2^5)^2 = 2^{5 \cdot 2} = 2^{10} $$
- Combine the Results
Putting both results together gives us:
$$ (y^3 \cdot 2^5)^2 = y^6 \cdot 2^{10} $$
The equivalent expression is $y^6 \cdot 2^{10}$.
More Information
This expression combines the results of raising each factor to the power of 2. Understanding the properties of exponents allows for easier manipulation and simplification.
Tips
- Forgetting to apply the exponent to each factor in a product.
- Miscalculating the exponent when multiplying the base and exponent, such as not multiplying correctly.
To avoid these mistakes, always ensure that you apply the exponent rules systematically to each individual term.
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