To simplify the expression \( 5^{10} \) using the Power Rule of Exponents, we can express it as follows:
The Power Rule states that \( (a^m)^n = a^{(m \cdot n)} \). However, in this case, we are interested in finding an expanded equivalent expression for \( 5^{10} \).
- The expression \( 5^{10} \) means multiplying \( 5 \) by itself \( 10 \) times: \[ 5^{10} = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \]
Now, let’s evaluate the provided options to see which one matches or is equivalent to \( 5^{10} \):
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Option 1: \( (5^{16})^6 \) \[ = 5^{(16 \cdot 6)} = 5^{96} \quad \text{(not equal to \( 5^{10} \))} \]
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Option 2: \( (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \cdot (5 \cdot 5 \cdot 5 \cdot 5 \cdot 5) \) \[ = (5^5) \cdot (5^5) \cdot (5^5) \cdot (5^5) \cdot (5^5) = 5^{(5+5+5+5+5)} = 5^{25} \quad \text{(not equal to \( 5^{10} \))} \]
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Option 3: \( (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \cdot (5 \cdot 5) \) \[ = (5^2) \cdot (5^2) \cdot (5^2) \cdot (5^2) \cdot (5^2) = 5^{(2+2+2+2+2)} = 5^{10} \quad \text{(this is equal to \( 5^{10} \))} \]
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Option 4: \( (5 \cdot 5)^8 \) \[ = (5^2)^8 = 5^{(2 \cdot 8)} = 5^{16} \quad \text{(not equal to \( 5^{10} \))} \]
From the above evaluation, only Option 3 is equal to \( 5^{10} \). Therefore, the correct answer is:
(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)