Question

which expression is equal to the expression below?
$(6 \cdot 9)^5$
a. $5 \cdot (6 \cdot 9)$
b. $15^5$
c. $6^5 \cdot 9^5$
d. $6^5 + 9^5$

Explanation:

Step1: Recall the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\) for any real numbers \(a\), \(b\) and positive integer \(n\).

Step2: Apply the rule to \((6\cdot9)^5\)

Using the power of a product rule with \(a = 6\), \(b=9\) and \(n = 5\), we get \((6\cdot9)^5=6^5\cdot9^5\).
We can also check the other options:

• Option A: \(5\cdot(6\cdot9)\) is a multiplication of 5 with the product of 6 and 9, which is not equal to \((6\cdot9)^5\) (since \((6\cdot9)^5=(6\cdot9)\times(6\cdot9)\times(6\cdot9)\times(6\cdot9)\times(6\cdot9)\) and \(5\cdot(6\cdot9)\) is just 5 times the product of 6 and 9).
• Option B: \(6 + 9=15\), but \((6\cdot9)^5=(54)^5\) and \(15^5\) is not equal to \(54^5\) (since \(54

eq15\)).

• Option D: \(6^5+9^5\) is the sum of two fifth - powers, while \((6\cdot9)^5\) is the fifth - power of a product, and they are not equal (for example, \(6^5 = 7776\), \(9^5=59049\), \(6^5 + 9^5=7776 + 59049=66825\); \((6\cdot9)^5=54^5 = 54\times54\times54\times54\times54=4627910496\), and \(66825

eq4627910496\)).

Answer:

C. \(6^{5}\cdot9^{5}\)