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Complete Question
Lacey and Haley are rewriting expressions in an equivalent, simpler form.
a. Haley simplified x³⋅ x² and got
x⁵
Lacey simplified x³ + x² and got the same result! However, their teacher told them that only one simplification is correct. Who simplified correctly and how do you know?
b. Haley simplifies 3⁵⋅ 4⁵ and gets the result 12^10, but Lacey is not sure.
Is Haley correct? Be sure to justify your answer.
Answer:
a) Haley is correct, Lacey simplified wrongly.
b) Haley is incorrect
Explanation:
a. Haley simplified x³⋅ x² and got
x⁵
Lacey simplified x³ + x² and got the same result! However, their teacher told them that only one simplification is correct. Who simplified correctly and how do you know?
For Question a, when it comes to simplifying algebraic expression that has to do with powers, there are certain rules that should be followed.
For example
x^a × x^b = x^(a + b)
For Haley, she simplified x³⋅ x² and got
x⁵
She is correct because this follows the product rule of powers or exponents above
= x³⋅ x² = x³+² = x⁵
For Lacey she is wrong because:
x³ + x² ≠ x⁵
x³ + x² when simplified as quadratic equation = x²(x + 1)
b. Haley simplifies 3⁵⋅ 4⁵ and gets the result 12^10, but Lacey is not sure.
Is Haley correct? Be sure to justify your answer.
For question b, when we have two distinct or different numbers with the same power(exponents) the rule states that:
x^a × y^a = (x × y)^a = (xy)^a
Haley is simplified wrongly. She did not apply the rule above
Haley simplified 3⁵⋅ 4⁵ = (3 × 4) ⁵+⁵
= 12^10, this is wrong.
The correct answer according to the rule =
3⁵⋅ 4⁵ = (3 × 4) ⁵ = 12⁵
Therefore,
3⁵⋅ 4⁵ ≠ 12^10
3⁵⋅ 4⁵ = 12⁵
Haley is wrong.