Question

do you understand?
1 essential question how do properties of integer exponents help you write equivalent expressions?

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?
2. construct arguments kristen wrote $5^8$ as an expression equivalent to $(5^2)^4$. her math partner writes $5^6$. who is correct?
3. critique reasoning tyler says that an equivalent expression for $2^3 \times 5^3$ is $10^9$. is he correct? explain.

do you know how?

1. write an equivalent expression for $7^{12} \cdot 7^4$.
2. write an equivalent expression for $(8^3)^4$.
3. a billboard has the given dimensions. using exponents, write two equivalent expressions for the area of the rectangle.
4. write an equivalent expression for $18^9 \div 18^1$.

Explanation:

Question 5

Step1: Recall exponent rule

When multiplying two powers with the same base, we use the rule \(a^m \cdot a^n = a^{m + n}\), where \(a = 7\), \(m = 12\), and \(n = 4\).

Step2: Apply the rule

For \(7^{12} \cdot 7^4\), add the exponents: \(12 + 4 = 16\). So the equivalent expression is \(7^{16}\).

Step1: Recall exponent rule

When raising a power to a power, we use the rule \((a^m)^n = a^{m \cdot n}\), where \(a = 8\), \(m = 3\), and \(n = 4\).

Step2: Apply the rule

For \((8^3)^4\), multiply the exponents: \(3 \times 4 = 12\). So the equivalent expression is \(8^{12}\).

Step1: Recall exponent rule

When dividing two powers with the same base, we use the rule \(a^m \div a^n = a^{m - n}\), where \(a = 18\), \(m = 9\), and \(n = 1\).

Step2: Apply the rule

For \(18^9 \div 18^1\), subtract the exponents: \(9 - 1 = 8\). So the equivalent expression is \(18^8\).

Answer:

\(7^{16}\)

Question 6