Question

properties of exponents: true or false
(question: i dont remember is this to think (this may variable))

1. $5^3 \cdot 5^2 = 5^5$ true false
2. $(4^2)^3 = 4^6$ true false
3. $\frac{6^5}{6^2} = 6^3$ true false
4. $\frac{8^7}{8^3} = 8^4$ true false
5. $4^3 \cdot 9^3 = 36^3$ true false
6. $1 = 10^0$ true false
7. $(8^3)^2 = 8^6$ true false
8. $9^{-2} = \frac{1}{9^2}$ true false
9. $9^7 \cdot 9^{-4} = 9^3$ true false
10. $\frac{5^3}{5^7} = 5^{-4}$ true false
11. $(7^3)^5 \cdot 7^6 = 7^{21}$ true false
12. $7^{\frac{1}{2}} \cdot 7^{-\frac{1}{2}} = 1^0$ true false
13. $\frac{(4^2)^5}{4^7} = 4^3$ true false
14. $\frac{15^0}{15^1} = \frac{1}{15}$ true false
15. $\frac{4^5 \cdot 4^7}{4^9} = 4^{-3}$ true false

Explanation:

To solve problems related to the properties of exponents (True or False), we use the following exponent rules:

Key Exponent Rules:

1. Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
2. Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
3. Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
4. Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)
5. Zero Exponent: \( a^0 = 1 \) (for \( a

eq 0 \))

Example Problem (e.g., First Box: \( 5^3 \cdot 5^2 = 5^5 \)):

Step 1: Apply Product of Powers Rule

The product of powers rule states \( a^m \cdot a^n = a^{m+n} \). For \( 5^3 \cdot 5^2 \), \( a = 5 \), \( m = 3 \), \( n = 2 \).
So, \( 5^3 \cdot 5^2 = 5^{3+2} = 5^5 \).

Step 2: Compare to Given Statement

The statement claims \( 5^3 \cdot 5^2 = 5^5 \), which matches the result from the rule.

Another Example (e.g., Box with \( \frac{8^5}{8^2} = 8^3 \)):

Step 1: Apply Quotient of Powers Rule

The quotient of powers rule states \( \frac{a^m}{a^n} = a^{m-n} \). For \( \frac{8^5}{8^2} \), \( a = 8 \), \( m = 5 \), \( n = 2 \).
So, \( \frac{8^5}{8^2} = 8^{5-2} = 8^3 \).

Step 2: Compare to Given Statement

The statement claims \( \frac{8^5}{8^2} = 8^3 \), which matches the result.

For a False Example (e.g., If a statement was \( (3^2)^4 = 3^6 \)):

Step 1: Apply Power of a Power Rule

The power of a power rule states \( (a^m)^n = a^{m \cdot n} \). For \( (3^2)^4 \), \( a = 3 \), \( m = 2 \), \( n = 4 \).
So, \( (3^2)^4 = 3^{2 \cdot 4} = 3^8 \).

Step 2: Compare to Given Statement

The statement claims \( (3^2)^4 = 3^6 \), but \( 3^8
eq 3^6 \), so the statement is False.

General Approach for Any Box:

1. Identify the exponent rule(s) relevant to the expression (product, quotient, power, negative, zero).
2. Simplify the left-hand side (LHS) using the rule(s).
3. Compare the simplified LHS to the right-hand side (RHS) of the statement.
4. If LHS = RHS, mark “True”; otherwise, mark “False.”

For each box, repeat this process using the appropriate exponent rule(s) to determine if the statement is True or False.

Answer:

To solve problems related to the properties of exponents (True or False), we use the following exponent rules:

Key Exponent Rules:

1. Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
2. Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
3. Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
4. Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)
5. Zero Exponent: \( a^0 = 1 \) (for \( a

eq 0 \))

Example Problem (e.g., First Box: \( 5^3 \cdot 5^2 = 5^5 \)):

Step 1: Apply Product of Powers Rule

The product of powers rule states \( a^m \cdot a^n = a^{m+n} \). For \( 5^3 \cdot 5^2 \), \( a = 5 \), \( m = 3 \), \( n = 2 \).
So, \( 5^3 \cdot 5^2 = 5^{3+2} = 5^5 \).

Step 2: Compare to Given Statement

The statement claims \( 5^3 \cdot 5^2 = 5^5 \), which matches the result from the rule.

Another Example (e.g., Box with \( \frac{8^5}{8^2} = 8^3 \)):

Step 1: Apply Quotient of Powers Rule

The quotient of powers rule states \( \frac{a^m}{a^n} = a^{m-n} \). For \( \frac{8^5}{8^2} \), \( a = 8 \), \( m = 5 \), \( n = 2 \).
So, \( \frac{8^5}{8^2} = 8^{5-2} = 8^3 \).

Step 2: Compare to Given Statement

The statement claims \( \frac{8^5}{8^2} = 8^3 \), which matches the result.

For a False Example (e.g., If a statement was \( (3^2)^4 = 3^6 \)):

Step 1: Apply Power of a Power Rule

The power of a power rule states \( (a^m)^n = a^{m \cdot n} \). For \( (3^2)^4 \), \( a = 3 \), \( m = 2 \), \( n = 4 \).
So, \( (3^2)^4 = 3^{2 \cdot 4} = 3^8 \).

Step 2: Compare to Given Statement

The statement claims \( (3^2)^4 = 3^6 \), but \( 3^8
eq 3^6 \), so the statement is False.

General Approach for Any Box:

1. Identify the exponent rule(s) relevant to the expression (product, quotient, power, negative, zero).
2. Simplify the left-hand side (LHS) using the rule(s).
3. Compare the simplified LHS to the right-hand side (RHS) of the statement.
4. If LHS = RHS, mark “True”; otherwise, mark “False.”

For each box, repeat this process using the appropriate exponent rule(s) to determine if the statement is True or False.