Question

1. look for relationships if you are writing an equivalent expression for $2^3 \cdot 2^4$, how many times would you write 2 as a factor?

1. construct arguments kristen wrote $5^8$ as an expression equivalent to $(5^2)^4$. her math partner writes $5^6$. who is correct?

1. critique reasoning tyler says that an equivalent expression for $2^3 \times 5^3$ is $10^9$. is he correct? explain.

Explanation:

Question 2

Step1: Recall exponent definition

$a^m$ means $a$ is multiplied $m$ times. So $2^3 = 2\times2\times2$ (3 times) and $2^4 = 2\times2\times2\times2$ (4 times).

Step2: Add the number of factors

To find the total number of times 2 is a factor in $2^3 \cdot 2^4$, we add the exponents (by the product of powers rule, $a^m \cdot a^n = a^{m + n}$). Here, $m = 3$, $n = 4$, so total factors of 2 is $3 + 4 = 7$.

Step1: Apply power of a power rule

The power of a power rule states that $(a^m)^n = a^{m\times n}$. For $(5^2)^4$, $m = 2$, $n = 4$, so $(5^2)^4 = 5^{2\times4}=5^8$.

Step2: Compare with the partner's answer

Kristen wrote $5^8$ and the partner wrote $5^6$. Since $5^8
eq5^6$, Kristen is correct.

Step1: Recall the product of powers with same exponent

The rule $a^m\times b^m=(a\times b)^m$. For $2^3\times5^3$, we can apply this rule: $2^3\times5^3=(2\times5)^3 = 10^3$, not $10^9$.

Step2: Explain Tyler's mistake

Tyler incorrectly assumed that we add the exponents, but the correct rule for multiplying powers with the same exponent is to multiply the bases and keep the exponent. So $2^3\times5^3 = 10^3$, so Tyler is not correct.

Answer:

7

Question 3