Answer:
8. Louis
9. Rose; Raymond
Step-by-step explanation:
8. An exponent represents the number of times the base appears as a factor in the product.
We use a coefficient to signify repeated addition: 3x means x+x+x.
We use an exponent to signify repeated multiplication. x³ means x·x·x.
So, the expression ...
[tex]4^2\cdot 4^5\text{ means }(4\cdot4)\cdot(4\cdot4\cdot4\cdot4\cdot4)[/tex]
You can see that the factor 4 appears 7 times in the product, so would be represented in exponential form as ...
[tex]4^2\cdot4^5=\boxed{4^7}[/tex]
Louis has correctly observed this fact. In general, we see that multiplying powers of the same base causes those powers to be added.
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9.
Part A. Rose is correct for the same reason as in problem 8.
5^5 · 5^2 = 5^(5+2) = 5^7
Part B. Raymond is correct. We know that division cancels similar terms from the numerator, so ...
[tex]\dfrac{7^9}{7^5}=\dfrac{7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7}{7\cdot7\cdot7\cdot7\cdot7}=7\cdot7\cdot7\cdot7=7^{9-5}=7^4[/tex]
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The rules of exponents we're using here are ...
(a^b)(a^c) = a^(b+c)
(a^b)/(a^c) = a^(b-c)
