Question

properties of exponents: true or false
determine whether each statement is true or false. circle your answer.

1. $3^2 \cdot 3^3 = 3^6$

true false

1. $(8^4)^2 = 8^6$

true false

1. $4^5 \div 4^2 = 4^3$

true false

1. $\frac{9^5}{9^2} = 9^3$

true false

1. $9^3 \cdot 9^4 = 9^{12}$

true false

1. $1^5 \cdot 1^3 = 1^8$

true false

1. $(3^2)^4 = 3^8$

true false

1. $9^0 = \frac{1}{9}$

true false

1. $9^3 \cdot 9^4 = 9^7$

true false

1. $\frac{8^{12}}{8^{11}} = 8^1$

true false

1. $(3^4)^3 \cdot 4^{10} = 3^{12}$

true false

1. $2^4 \cdot 2^3 = 2^{12}$

true false

1. $(\frac{4^7}{4^3})^2 = 4^8$

true false

1. $\frac{13^5}{13^7} = \frac{1}{13^2}$

true false

1. $\frac{4^9 \cdot 4^{15}}{4^7} = 4^{16}$

true false
challenge!
correct all of the false statements. change one side of each false equation to make it true.

Explanation:

Let's take the first problem: \( 3^2 \cdot 3^4 = 3^6 \)

Step1: Recall exponent rule

When multiplying exponents with the same base, we add the exponents: \( a^m \cdot a^n = a^{m + n} \)

Step2: Apply the rule

For \( 3^2 \cdot 3^4 \), \( m = 2 \), \( n = 4 \), so \( 3^{2 + 4}=3^6 \)
So the left - hand side \( 3^2\cdot3^4 = 3^{2 + 4}=3^6 \), which is equal to the right - hand side. So this statement is True.

Let's take the second problem: \( (8^3)^2=8^5 \)

Step1: Recall exponent rule

When raising a power to a power, we multiply the exponents: \( (a^m)^n=a^{m\times n} \)

Step2: Apply the rule

For \( (8^3)^2 \), \( m = 3 \), \( n = 2 \), so \( 8^{3\times2}=8^6
eq8^5 \)
So this statement is False.

(We can continue this process for each of the problems. For example, for \( \frac{9^5}{9^2}=9^3 \))

Step1: Recall exponent rule

When dividing exponents with the same base, we subtract the exponents: \( \frac{a^m}{a^n}=a^{m - n} \)

Step2: Apply the rule

For \( \frac{9^5}{9^2} \), \( m = 5 \), \( n = 2 \), so \( 9^{5 - 2}=9^3 \), so this statement is True.

(We will go through each problem one by one using the exponent rules: \( a^m\cdot a^n=a^{m + n} \), \( (a^m)^n=a^{mn} \), \( \frac{a^m}{a^n}=a^{m - n} \), \( a^0 = 1(a
eq0) \), \( a^{-n}=\frac{1}{a^n} \))

Answer:

1. \( 3^2 \cdot 3^4 = 3^6 \): True
2. \( (8^3)^2 = 8^5 \): False
3. \( 4^7=4^5 \): False (since \( 7

eq5 \))

1. \( \frac{9^5}{9^2}=9^3 \): True
2. \( 8^2\cdot8^3 = 8^6 \): False (\( 8^{2 + 3}=8^5

eq8^6 \))

1. \( 1 = 4^0 \): True (by the rule \( a^0 = 1,a

eq0 \))

1. \( (3^4)^2=3^8 \): True (\( (a^m)^n=a^{mn} \), \( 3^{4\times2}=3^8 \))
2. \( 9^{-1}=\frac{1}{9^1} \): True (by the rule \( a^{-n}=\frac{1}{a^n} \))
3. \( 9^3\cdot9^4 = 9^7 \): True (\( 9^{3+4}=9^7 \))
4. \( \frac{8^{12}}{8^{11}}=8^1 \): True (\( 8^{12 - 11}=8^1 \))
5. \( (3^4)^3\cdot4^0 = 3^{12} \): True (\( (3^4)^3=3^{12} \), \( 4^0 = 1 \), so \( 3^{12}\times1=3^{12} \))
6. \( 8^{-4}\cdot8^4 = 8^0 \): True (\( 8^{-4 + 4}=8^0 \))
7. \( (\frac{4^7}{4^7})^3=4^0 \): True (\( \frac{4^7}{4^7}=1 = 4^0 \), \( (4^0)^3=4^0 \))
8. \( (\frac{1}{13})^2=\frac{1}{13^2} \): True (by the rule \( (\frac{1}{a})^n=\frac{1}{a^n} \))
9. \( \frac{4^9\cdot4^{15}}{4^7}=4^{17} \): True (\( 4^{9 + 15-7}=4^{17} \))

(Note: For each false statement, we can also rewrite it to make it true. For example, \( (8^3)^2 = 8^6 \) instead of \( (8^3)^2 = 8^5 \), \( 4^7=4^7 \) or \( 4^7 = 4^{5+2}=4^5\times4^2 \) instead of \( 4^7 = 4^5 \), \( 8^2\cdot8^3 = 8^5 \) instead of \( 8^2\cdot8^3 = 8^6 \))