Asked by UnicornSparkles
Which expression is equivalent to 6^7/6^8x6-^7
Answers
Answered by
UnicornSparkles
Which expression is equivalent to 4^2/4^3•4^0
Answered by
UnicornSparkles
Which expression is equivalent to (3-^1/3^6) -^5
Answered by
UnicornSparkles
Which expression is equivalent to (2-^0/2)^2
Answered by
UnicornSparkles
Which expression is equivalent to (6^0)^4/6^5
Answered by
UnicornSparkles
Which expression is equivalent to 5-^1/5-^2 • 5-^3
Answered by
GPT-4o mini
To simplify the expression \( \frac{6^7}{6^8 \times 6^{-7}} \), we can use the properties of exponents.
1. First, we can combine the exponents in the denominator using the property \( a^m \times a^n = a^{m+n} \):
\[
6^8 \times 6^{-7} = 6^{8 + (-7)} = 6^{8 - 7} = 6^{1} = 6
\]
2. Now the expression simplifies to:
\[
\frac{6^7}{6}
\]
3. We can apply the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{6^7}{6^1} = 6^{7 - 1} = 6^6
\]
Thus, the expression \( \frac{6^7}{6^8 \times 6^{-7}} \) simplifies to \( 6^6 \).
So, the equivalent expression is \( 6^6 \).
1. First, we can combine the exponents in the denominator using the property \( a^m \times a^n = a^{m+n} \):
\[
6^8 \times 6^{-7} = 6^{8 + (-7)} = 6^{8 - 7} = 6^{1} = 6
\]
2. Now the expression simplifies to:
\[
\frac{6^7}{6}
\]
3. We can apply the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{6^7}{6^1} = 6^{7 - 1} = 6^6
\]
Thus, the expression \( \frac{6^7}{6^8 \times 6^{-7}} \) simplifies to \( 6^6 \).
So, the equivalent expression is \( 6^6 \).
Answered by
GPT-4o mini
To simplify the expression \( \frac{4^2}{4^3 \cdot 4^0} \), we can use the properties of exponents.
1. First, simplify \( 4^3 \cdot 4^0 \) using the property \( a^m \cdot a^n = a^{m+n} \):
\[
4^3 \cdot 4^0 = 4^{3 + 0} = 4^3
\]
2. Now the expression simplifies to:
\[
\frac{4^2}{4^3}
\]
3. Next, we can apply the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{4^2}{4^3} = 4^{2 - 3} = 4^{-1}
\]
4. Finally, we can express \( 4^{-1} \) as:
\[
4^{-1} = \frac{1}{4}
\]
Thus, the expression \( \frac{4^2}{4^3 \cdot 4^0} \) simplifies to \( 4^{-1} \) or \( \frac{1}{4} \).
So, the equivalent expression is \( \frac{1}{4} \).
1. First, simplify \( 4^3 \cdot 4^0 \) using the property \( a^m \cdot a^n = a^{m+n} \):
\[
4^3 \cdot 4^0 = 4^{3 + 0} = 4^3
\]
2. Now the expression simplifies to:
\[
\frac{4^2}{4^3}
\]
3. Next, we can apply the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{4^2}{4^3} = 4^{2 - 3} = 4^{-1}
\]
4. Finally, we can express \( 4^{-1} \) as:
\[
4^{-1} = \frac{1}{4}
\]
Thus, the expression \( \frac{4^2}{4^3 \cdot 4^0} \) simplifies to \( 4^{-1} \) or \( \frac{1}{4} \).
So, the equivalent expression is \( \frac{1}{4} \).
Answered by
GPT-4o mini
To simplify the expression \( \left( \frac{3^{-1}}{3^6} \right)^{-5} \), we can follow these steps:
1. **Simplify the fraction inside the parentheses:**
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify \(\frac{3^{-1}}{3^6}\):
\[
\frac{3^{-1}}{3^6} = 3^{-1 - 6} = 3^{-7}
\]
2. **Now replace the fraction in the original expression:**
The expression now looks like:
\[
(3^{-7})^{-5}
\]
3. **Apply the power of a power property:**
Using the property \((a^m)^n = a^{m \cdot n}\):
\[
(3^{-7})^{-5} = 3^{-7 \cdot -5} = 3^{35}
\]
Thus, the expression \( \left( \frac{3^{-1}}{3^6} \right)^{-5} \) simplifies to \( 3^{35} \).
The equivalent expression is \( 3^{35} \).
1. **Simplify the fraction inside the parentheses:**
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify \(\frac{3^{-1}}{3^6}\):
\[
\frac{3^{-1}}{3^6} = 3^{-1 - 6} = 3^{-7}
\]
2. **Now replace the fraction in the original expression:**
The expression now looks like:
\[
(3^{-7})^{-5}
\]
3. **Apply the power of a power property:**
Using the property \((a^m)^n = a^{m \cdot n}\):
\[
(3^{-7})^{-5} = 3^{-7 \cdot -5} = 3^{35}
\]
Thus, the expression \( \left( \frac{3^{-1}}{3^6} \right)^{-5} \) simplifies to \( 3^{35} \).
The equivalent expression is \( 3^{35} \).
Answered by
GPT-4o mini
To simplify the expression \( \left( \frac{2^0}{2} \right)^2 \), we can follow these steps:
1. **Evaluate \(2^0\):**
By the property of exponents, any non-zero number raised to the power of 0 is 1:
\[
2^0 = 1
\]
2. **Substitute into the fraction:**
Now substitute \(2^0\) into the expression:
\[
\frac{2^0}{2} = \frac{1}{2}
\]
3. **Now, replace the fraction in the original expression:**
The expression now looks like:
\[
\left( \frac{1}{2} \right)^2
\]
4. **Square the fraction:**
Squaring the fraction results in:
\[
\left( \frac{1}{2} \right)^2 = \frac{1^2}{2^2} = \frac{1}{4}
\]
Thus, the expression \( \left( \frac{2^0}{2} \right)^2 \) simplifies to \( \frac{1}{4} \).
The equivalent expression is \( \frac{1}{4} \).
1. **Evaluate \(2^0\):**
By the property of exponents, any non-zero number raised to the power of 0 is 1:
\[
2^0 = 1
\]
2. **Substitute into the fraction:**
Now substitute \(2^0\) into the expression:
\[
\frac{2^0}{2} = \frac{1}{2}
\]
3. **Now, replace the fraction in the original expression:**
The expression now looks like:
\[
\left( \frac{1}{2} \right)^2
\]
4. **Square the fraction:**
Squaring the fraction results in:
\[
\left( \frac{1}{2} \right)^2 = \frac{1^2}{2^2} = \frac{1}{4}
\]
Thus, the expression \( \left( \frac{2^0}{2} \right)^2 \) simplifies to \( \frac{1}{4} \).
The equivalent expression is \( \frac{1}{4} \).
Answered by
GPT-4o mini
To simplify the expression \( \frac{(6^0)^4}{6^5} \), we can follow these steps:
1. **Evaluate \(6^0\):**
By the property of exponents, any non-zero number raised to the power of 0 is 1:
\[
6^0 = 1
\]
2. **Replace \(6^0\) in the expression:**
The expression now looks like:
\[
(6^0)^4 = 1^4 = 1
\]
3. **Substitute back into the fraction:**
The expression simplifies to:
\[
\frac{1}{6^5}
\]
Thus, the expression \( \frac{(6^0)^4}{6^5} \) simplifies to \( \frac{1}{6^5} \).
The equivalent expression is \( \frac{1}{6^5} \).
1. **Evaluate \(6^0\):**
By the property of exponents, any non-zero number raised to the power of 0 is 1:
\[
6^0 = 1
\]
2. **Replace \(6^0\) in the expression:**
The expression now looks like:
\[
(6^0)^4 = 1^4 = 1
\]
3. **Substitute back into the fraction:**
The expression simplifies to:
\[
\frac{1}{6^5}
\]
Thus, the expression \( \frac{(6^0)^4}{6^5} \) simplifies to \( \frac{1}{6^5} \).
The equivalent expression is \( \frac{1}{6^5} \).
Answered by
GPT-4o mini
To simplify the expression \( \frac{5^{-1}}{5^{-2} \cdot 5^{-3}} \), we can use the properties of exponents.
1. **Simplify the denominator:**
Use the property \( a^m \cdot a^n = a^{m+n} \) to combine the powers in the denominator:
\[
5^{-2} \cdot 5^{-3} = 5^{-2 + (-3)} = 5^{-2 - 3} = 5^{-5}
\]
2. **Now rewrite the expression:**
The expression now looks like:
\[
\frac{5^{-1}}{5^{-5}}
\]
3. **Apply the property of exponents:**
Use the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{5^{-1}}{5^{-5}} = 5^{-1 - (-5)} = 5^{-1 + 5} = 5^{4}
\]
Hence, the simplified expression \( \frac{5^{-1}}{5^{-2} \cdot 5^{-3}} \) is equivalent to \( 5^4 \).
1. **Simplify the denominator:**
Use the property \( a^m \cdot a^n = a^{m+n} \) to combine the powers in the denominator:
\[
5^{-2} \cdot 5^{-3} = 5^{-2 + (-3)} = 5^{-2 - 3} = 5^{-5}
\]
2. **Now rewrite the expression:**
The expression now looks like:
\[
\frac{5^{-1}}{5^{-5}}
\]
3. **Apply the property of exponents:**
Use the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{5^{-1}}{5^{-5}} = 5^{-1 - (-5)} = 5^{-1 + 5} = 5^{4}
\]
Hence, the simplified expression \( \frac{5^{-1}}{5^{-2} \cdot 5^{-3}} \) is equivalent to \( 5^4 \).
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