Sure, let's match each product of powers with their simplified expression step-by-step:
1. For the expression \(5^{-4} \cdot 5^4 \cdot 5^0\):
- Using the properties of exponents, we can add the exponents:
\(-4 + 4 + 0 = 0\).
- Therefore, the simplified form of \(5^{-4} \cdot 5^4 \cdot 5^0\) is:
[tex]\[ 5^0 = 1. \][/tex]
- Matched expression: \((1.0, 1)\).
2. For the expression \(5^6 \cdot 5^{-4}\):
- Using the properties of exponents, we can add the exponents:
\(6 - 4 = 2\).
- Therefore, the simplified form of \(5^6 \cdot 5^{-4}\) is:
[tex]\[ 5^2 = 25. \][/tex]
- Matched expression: \((25.0, 25)\).
3. For the expression \(5^{-3} \cdot 5^{-3}\):
- Using the properties of exponents, we can add the exponents:
\(-3 + -3 = -6\).
- Therefore, the simplified form of \(5^{-3} \cdot 5^{-3}\) is:
[tex]\[ 5^{-6} = \frac{1}{5^6} \approx 6.4 \times 10^{-5}. \][/tex]
- Matched expression: \((6.4e-05, 6.4e-05)\).
4. For the expression \(5 \cdot 5^3\):
- Using the properties of exponents, we can add the exponents:
\(1 + 3 = 4\).
- Therefore, the simplified form of \(5 \cdot 5^3\) is:
[tex]\[ 5^4 = 625. \][/tex]
- Matched expression: \((625, 625)\).
5. For the expression \(5^7 \cdot 5^3\):
- Using the properties of exponents, we can add the exponents:
\(7 + 3 = 10\).
- Therefore, the simplified form of \(5^7 \cdot 5^3\) is:
[tex]\[ 5^{10} = 9,765,625. \][/tex]
- Matched expression: \((9765625, 9765625)\).
To summarize:
- \(5^{-4} \cdot 5^4 \cdot 5^0 \rightarrow 5^0 = 1\)
- \(5^6 \cdot 5^{-4} \rightarrow 5^2 = 25\)
- \(5^{-3} \cdot 5^{-3} \rightarrow 5^{-6} = \frac{1}{5^6} = 6.4 \times 10^{-5}\)
- \(5 \cdot 5^3 \rightarrow 5^4 = 625\)
- [tex]\(5^7 \cdot 5^3 \rightarrow 5^{10} = 9,765,625\)[/tex]
