Question

1. critique reasoning alberto incorrectly stated that $\frac{8^{7}}{8^{2}} = 1$. what was albertos error? explain your reasoning and find the correct answer.
2. is the expression $8\times8^{5}$ equivalent to $(8\times8)^{5}$? explain.
3. is the expression $(3^{7})^{-3}$ equivalent to $(3^{3})^{-7}$? explain.
4. is the expression $3^{2}cdot3^{-3}$ equivalent to $3^{-1}$? explain.
5. model with math what is the width of the rectangle written as an exponential expression? area = $10^{4} m^{2}$, $10^{3} m$
6. simplify the expression $((\frac{1}{2})^{3})^{3}$.
7. higher order thinking use a property of exponents to write $(3b)^{5}$ as a product of powers.
8. select all the expressions equivalent to $4^{5}cdot4^{10}$. $4^{5}+4^{10}$, $4^{3}cdot4^{5}$, $4^{3}cdot4^{12}$, $4^{3}+4^{12}$, $4^{18}-4^{3}$, $4^{15}$
9. your teacher asks the class to evaluate the expression $(2^{3})^{1}$. your classmate gives an incorrect answer of 16. part a evaluate the expression. part b what was the likely error? a your classmate divided the exponents. b your classmate multiplied the exponents. c your classmate added the exponents. d your classmate subtracted the exponents.

1 - 6 use properties of integer exponents

Explanation:

Step1: Recall exponent - multiplication rule

When multiplying two numbers with the same base \(a^m\times a^n=a^{m + n}\). For \(4^5\times4^{10}\), using the rule, we have \(4^{5+10}=4^{15}\).

Step2: Analyze each option

• For \(4^5 + 4^{10}\), addition of exponential terms with the same base is not the same as multiplication, so \(4^5+4^{10}

eq4^5\times4^{10}\).

• For \(4^3\times4^5=4^{3 + 5}=4^8

eq4^5\times4^{10}\).

• For \(4^3\times4^{12}=4^{3+12}=4^{15}\), which is equivalent to \(4^5\times4^{10}\).
• For \(4^3 + 4^{12}\), addition of exponential terms with the same base is not the same as multiplication, so \(4^3+4^{12}

eq4^5\times4^{10}\).

• For \(4^{18}-4^3\), subtraction of exponential terms with the same base is not the same as multiplication, so \(4^{18}-4^3

eq4^5\times4^{10}\).

• For \(4^{15}\), we already found that \(4^5\times4^{10}=4^{15}\).

Step3: Evaluate \((2^3)^1\)

Using the power - of - a - power rule \((a^m)^n=a^{mn}\), for \((2^3)^1\), we have \(2^{3\times1}=2^3 = 8\).

Step4: Identify the error

The classmate likely made an error in applying the exponent rules. The correct rule for \((a^m)^n\) is to multiply the exponents. If the classmate got 16, they might have thought \(2^{3 + 1}=2^4=16\), so they added the exponents.

Answer:

1. C. \(4^3\cdot4^{12}\), F. \(4^{15}\)
2. PART A: 8

PART B: C. Your classmate added the exponents.