Which expression is equivalent to (5ab^3)² for all values of a and b where the expression is defined?
Understand the Problem
The question is asking to find an expression that is equivalent to \( (5ab^3)^{2} \) for all values of a and b where the expression is defined. This requires understanding algebraic expressions and their equivalences.
Answer
The expression equivalent to \( (5ab^3)^{2} \) is \( 25a^2b^6 \).
Answer for screen readers
The expression equivalent to ( (5ab^3)^{2} ) is ( 25a^2b^6 ).
Steps to Solve
- Identify the expression to be simplified
We start with the expression ( (5ab^3)^{2} ).
- Apply the exponent to each component
According to the power of a product rule, we will apply the exponent (2) to each part of the expression:
[ (5ab^3)^2 = 5^2 \cdot (a)^2 \cdot (b^3)^2 ]
- Calculate (5^2)
Calculating (5^2): [ 5^2 = 25 ]
- Calculate ((b^3)^2)
Using the power of a power rule, we can simplify ((b^3)^2): [ (b^3)^2 = b^{3 \times 2} = b^6 ]
- Combine the results into one expression
Now we can rewrite the expression by substituting back the computed values: [ (5ab^3)^2 = 25 \cdot a^2 \cdot b^6 ]
- Final expression
The final equivalent expression is: [ 25a^2b^6 ]
The expression equivalent to ( (5ab^3)^{2} ) is ( 25a^2b^6 ).
More Information
When simplifying expressions, it's important to apply algebraic rules carefully, like the power of a product and power of a power. This leads to more manageable forms for mathematical expressions.
Tips
- Forgetting to apply the exponent to all components of the product.
- Miscalculating the powers (e.g., ( (b^3)^2 )).
- Not keeping track of coefficients when simplifying.
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