Question

properties of exponents: true or false
determine if each equation is true or false. circle your answer.

1. $3^2 \cdot 3^3 = 3^5$

true false

1. $(8^3)^2 = 8^5$

true false

1. $\frac{4^5}{4^2} = 4^3$

true false

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

true false

1. $8^3 \cdot 8^7 = 8^{10}$

true false

1. $1 = 2^0$

true false

1. $(2^3)^4 = 2^{12}$

true false

1. $5^{-2} = \frac{1}{5^2}$

true false

1. $8^3 \cdot 8^{-2} = 8^5$

true false

1. $\frac{6^{12}}{6^7} = 6^5$

true false

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

true false

1. $x^{-\frac{1}{2}} \cdot x^{\frac{1}{2}} = x^0$

true false

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

true false

1. $\frac{12^0}{12^3} = \frac{1}{12^3}$

true false

1. $\frac{4^m \cdot 4^n}{4^p} = 4^{m + n - p}$

true false

Explanation:

Let's take the first problem: \(3^2 \cdot 3^3 = 3^5\)

Step 1: Recall the exponent rule for multiplication

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

Step 2: Apply the rule to the left - hand side

For \(3^2 \cdot 3^3\), here \(a = 3\), \(m=2\) and \(n = 3\). So \(3^2\cdot3^3=3^{2 + 3}=3^5\)
The left - hand side is equal to the right - hand side, so this statement is True.

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

Step 1: Recall the exponent rule for power of a power

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

Step 2: Apply the rule to the left - hand side

For \((8^2)^3\), \(a = 8\), \(m = 2\) and \(n=3\). So \((8^2)^3=8^{2\times3}=8^6
eq8^5\)
So this statement is False.

Let's take the third problem: \(\frac{4^7}{4^3}=4^4\)

Step 1: Recall the exponent rule for division

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

Step 2: Apply the rule to the left - hand side

For \(\frac{4^7}{4^3}\), \(a = 4\), \(m = 7\) and \(n=3\). So \(\frac{4^7}{4^3}=4^{7 - 3}=4^4\)
The left - hand side is equal to the right - hand side, so this statement is True.

(We can continue this process for each of the problems using the relevant exponent rules:

• \(a^m\cdot a^n=a^{m + n}\) (product of powers)
• \((a^m)^n=a^{m\times n}\) (power of a power)
• \(\frac{a^m}{a^n}=a^{m - n}\) (quotient of powers)
• \(a^0 = 1\) (zero exponent)
• \(a^{-n}=\frac{1}{a^n}\) (negative exponent)

)

Answer:

For each problem:

1. \(3^2\cdot3^3 = 3^5\): True (using \(a^m\cdot a^n=a^{m + n}\), \(2 + 3=5\))
2. \((8^2)^3=8^5\): False (using \((a^m)^n=a^{m\times n}\), \(2\times3 = 6

eq5\))

1. \(\frac{4^7}{4^3}=4^4\): True (using \(\frac{a^m}{a^n}=a^{m - n}\), \(7-3 = 4\))
2. \(\frac{8^5}{8^3}=8^2\): True (using \(\frac{a^m}{a^n}=a^{m - n}\), \(5 - 3=2\))
3. \(8^3\cdot8^7=8^{10}\): True (using \(a^m\cdot a^n=a^{m + n}\), \(3+7 = 10\))
4. \(1 = 2^0\): True (using \(a^0 = 1\) for \(a = 2\))
5. \((2^3)^5=2^{15}\): True (using \((a^m)^n=a^{m\times n}\), \(3\times5=15\))
6. \(5^{-2}=\frac{1}{5^2}\): True (using \(a^{-n}=\frac{1}{a^n}\))
7. \(8^3\cdot8^{-2}=8^1\): True (using \(a^m\cdot a^n=a^{m + n}\), \(3+( - 2)=1\))
8. \(\frac{6^{12}}{6^7}=6^5\): True (using \(\frac{a^m}{a^n}=a^{m - n}\), \(12-7 = 5\))
9. \((y^4)\cdot y^5=y^9\): True (using \(a^m\cdot a^n=a^{m + n}\), \(4 + 5=9\))
10. \(x^{-\frac{1}{2}}x^{\frac{1}{2}}=x^0\): True (using \(a^m\cdot a^n=a^{m + n}\), \(-\frac{1}{2}+\frac{1}{2}=0\) and \(x^0 = 1\))
11. \(\frac{(4^3)^2}{4^2}=4^4\): True (First, \((4^3)^2=4^{3\times2}=4^6\), then \(\frac{4^6}{4^2}=4^{6 - 2}=4^4\))
12. \(\frac{12^0}{12^2}=\frac{1}{12^2}\): True (Since \(12^0 = 1\), \(\frac{1}{12^2}=\frac{12^0}{12^2}\) and using \(\frac{a^m}{a^n}=a^{m - n}\), \(0-2=-2\) and \(a^{-n}=\frac{1}{a^n}\))
13. \(\frac{4^m\cdot4^n}{4^p}=4^{m + n-p}\): True (Using \(a^m\cdot a^n=a^{m + n}\) for the numerator, we get \(4^{m + n}\), then using \(\frac{a^m}{a^n}=a^{m - n}\), \(\frac{4^{m + n}}{4^p}=4^{(m + n)-p}=4^{m + n - p}\))