Testinar Grade 8 math

Properties of Integer Exponents

Learn to reason about and solve Grade 8 problems involving Properties of Integer Exponents.

CH02-01 Exponents, Roots, and Scientific Notation

Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

Which pair of expressions is equivalent?
1. Option A
2. Option B
3. Option C
4. Option D
Which expression is equivalent to \(\frac{5^6}{5^2}\)?
1. Option D
2. Option B
3. Option C
4. Option A
Look at the table. Which expression is not equivalent to \(x^6\)?
1. Option A
2. Option B
3. Option C
4. Option D
Match the expression to the exponent law it uses: Which expression uses the quotient rule?
1. \(a^2 \cdot a^4 = a^6\)
2. \((y^2)^5 = y^{10}\)
3. \((2b)^3 = 8b^3\)
4. \(\frac{m^9}{m^3} = m^6\)
Visual: Expanded form. \(2^5\) means: If we multiply \(2^3 \times 2^4\), how many total factors of 2 do we have?
1. \(2\)
2. \(12\)
3. \(24\)
4. \(7\)
Complete the table by applying exponent laws: What is the exponent in the result?
1. \(3\)
2. \(10\)
3. \(15\)
4. \(7\)
Using the quotient rule, explain why \(x^0 = 1\) (for \(x \neq 0\)). Since \(\frac{x^5}{x^5} = 1\), what must \(x^0\) equal?
1. Because the exponent is 0, so the result is 0.
2. Because the quotient rule says so.
3. Because when exponents are equal, the quotient is 0.
4. \(1\) (any nonzero number divided by itself equals 1)
Which law does this diagram represent? Which of the following uses this same law?
1. \((5a)^2 = 25a^2\)
2. \(\frac{p^7}{p^3} = p^4\)
3. \(2^4 \cdot 2^5 = 2^9\)
4. \((y^3)^4 = y^{12}\)

Teacher and parent reference

Answer key and explanations

C. Option C
Check each: (A) \(x^5 \neq x^6\); (B) \(x^6 \neq x^5\); (C) \(x^4 = x^4\) ✓; (D) addition, not a law.
D. Option A
\(\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625\).
D. Option D
(D) is \(x^6 + 1\), not \(x^6\), because addition is not an exponent law.
D. \(\frac{m^9}{m^3} = m^6\)
Quotient rule subtracts exponents: \(\frac{m^9}{m^3} = m^{9-3} = m^6\). (A) is product, (B) is power of power, (C) is power of product.
D. \(7\)
\(2^3\) has 3 factors, \(2^4\) has 4 factors. Total: \(3 + 4 = 7\) factors. This illustrates the product rule: \(2^3 \times 2^4 = 2^7\).
D. \(7\)
Product rule: \(3^5 \cdot 3^2 = 3^{5+2} = 3^7\).
D. \(1\) (any nonzero number divided by itself equals 1)
The quotient rule shows: \(\frac{x^5}{x^5} = x^{5-5} = x^0\). Since \(\frac{x^5}{x^5} = 1\), we have \(x^0 = 1\).
C. \(2^4 \cdot 2^5 = 2^9\)
The diagram shows the product rule (add exponents). (C) uses it: \(2^{4+5} = 2^9\). (A) is power of product, (B) is quotient rule, (D) is power of power.