To simplify \( 5^{10} \) using the Power Rule of Exponents, we start by understanding what \( 5^{10} \) represents. The expression \( 5^{10} \) means that 5 is multiplied by itself a total of 10 times.
Now, let's analyze the options given:
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\( (5^{16})^6 \): By the power rule, \( (a^m)^n = a^{m \cdot n} \), thus: \[ (5^{16})^6 = 5^{16 \cdot 6} = 5^{96} \] This does not simplify to \( 5^{10} \).
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\( (5⋅5⋅5⋅5⋅5)⋅(5⋅5⋅5⋅5⋅5)⋅(5⋅5⋅5⋅5⋅5)⋅(5⋅5⋅5⋅5⋅5)⋅(5⋅5⋅5⋅5⋅5) \): This expression can be counted as 5 groups of 5, where each group has 5 factors of 5: \[ (5 \times 5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5 \times 5 \times 5) \] This equals \( 5^{5 + 5 + 5 + 5 + 5} = 5^{25} \). This does not simplify to \( 5^{10} \).
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\( (5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5) \): Each \( (5 \times 5) \) represents \( 5^2 \). Therefore: \[ (5 \times 5) \times (5 \times 5) \times (5 \times 5) \times (5 \times 5) \times (5 \times 5) = 5^2 \times 5^2 \times 5^2 \times 5^2 \times 5^2 \] By adding the exponents, this becomes \( 5^{2 + 2 + 2 + 2 + 2} = 5^{10} \). This is equivalent to \( 5^{10} \).
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\( (5⋅5)^8 \): This means \( (5^2)^8 \). By the power rule: \[ (5^2)^8 = 5^{2 \cdot 8} = 5^{16} \] This does not simplify to \( 5^{10} \).
From the above analysis, the correct expanded equivalent expression for \( 5^{10} \) is:
\( (5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5)⋅(5⋅5) \).
2 answers