Question

do you understand?

1. essential question how do properties of integer exponents help you write equivalent expressions?
2. 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?
3. construct arguments kristen wrote (5^8) as an expression equivalent to ((5^2)^4). her math partner writes (5^6). who is correct?
4. 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 rule

When multiplying exponents with the same base, we use the rule \(a^m \cdot a^n = a^{m + n}\). For \(2^3 \cdot 2^4\), the base \(a = 2\), \(m = 3\), \(n = 4\).

Step2: Calculate the sum of exponents

First, find the sum of the exponents: \(3 + 4 = 7\). So \(2^3 \cdot 2^4 = 2^{3 + 4}=2^7\).

Step3: Interpret the exponent

The exponent of 7 means we write 2 as a factor 7 times (since \(2^7 = 2\times2\times2\times2\times2\times2\times2\)).

Step1: Recall power of a power rule

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

Step2: Apply the rule

Calculate \(m\times n\): \(2\times4 = 8\). So \((5^2)^4 = 5^{2\times4}=5^8\).

Step3: Compare with the partner's answer

Kristen wrote \(5^8\) and the partner wrote \(5^6\). Since \((5^2)^4 = 5^8\), Kristen is correct.

Step1: Recall product of powers rule for different bases

The rule \((ab)^n=a^n\times b^n\) (reverse of power of a product). For \(2^3\times5^3\), we can rewrite it as \((2\times5)^3\).

Step2: Simplify the base

Calculate \(2\times5 = 10\), so \((2\times5)^3 = 10^3\).

Step3: Compare with Tyler's answer

Tyler says it's \(10^9\), but we found it's \(10^3\). So Tyler is not correct.

Answer:

7

Question 3