Question

elisa says ((7 cdot 15)^3) is equivalent to (7^6 cdot 15^6). her work is shown below.
is elisa correct? if not, how can she correct her error?
yes, elisa is correct.
no, step 1 should be (3 cdot 7 cdot 15).
no, step 2 should be (7 cdot 15^6).
no, step 3 should be (7^3 cdot 15^3).
step 1 ((7 cdot 15)^3 = (7 cdot 15) cdot (7 cdot 15) cdot (7 cdot 15)
step 2 quadquadquadquadquad= 7 cdot 7 cdot 7 cdot 15 cdot 15 cdot 15
step 3 quadquadquadquadquad= 7^6 cdot 15^6)

Explanation:

Step1: Recall exponent rule for products

The power of a product rule states that \((ab)^n = a^n b^n\), where \(a\) and \(b\) are real numbers and \(n\) is a positive integer. This means when we raise a product to a power, we raise each factor to that power.

Step2: Analyze Elisa's steps

• Step 1: \((7\cdot15)^3=(7\cdot15)\cdot(7\cdot15)\cdot(7\cdot15)\) is correct because \(a^3=a\cdot a\cdot a\) for any \(a\).
• Step 2: \((7\cdot15)\cdot(7\cdot15)\cdot(7\cdot15)=7\cdot7\cdot7\cdot15\cdot15\cdot15\) is correct by the commutative and associative properties of multiplication (we can rearrange and group the factors).
• Step 3: Elisa claims \(7\cdot7\cdot7\cdot15\cdot15\cdot15 = 7^6\cdot15^6\), but this is incorrect. Since \(7\cdot7\cdot7=7^3\) (because we have 3 factors of 7) and \(15\cdot15\cdot15 = 15^3\) (because we have 3 factors of 15), the correct simplification is \(7^3\cdot15^3\).

Answer:

No, Step 3 should be \(7^{3}\cdot15^{3}\). (The option corresponding to this is the last one: "No, Step 3 should be \(7^{3}\cdot15^{3}\).")