To solve the equation [tex]\((\sqrt{5})^7 \div (\sqrt{5})^5 = 5^x\)[/tex], we need to simplify the left-hand side and then equate it to the right-hand side.
1. Simplify the left-hand side using the rules of exponents:
[tex]\[(\sqrt{5})^7 \div (\sqrt{5})^5\][/tex]
2. Recall the property of exponents:
[tex]\[\frac{a^m}{a^n} = a^{m-n}\][/tex]
Applying this property to our equation:
[tex]\[ (\sqrt{5})^7 \div (\sqrt{5})^5 = (\sqrt{5})^{7-5} = (\sqrt{5})^2 \][/tex]
3. Simplify [tex]\((\sqrt{5})^2\)[/tex]:
[tex]\[ (\sqrt{5})^2 = \sqrt{5} \times \sqrt{5} = 5 \][/tex]
This means:
[tex]\[ (\sqrt{5})^7 \div (\sqrt{5})^5 = 5 \][/tex]
4. Now equate this to the right-hand side:
[tex]\[ 5 = 5^x \][/tex]
5. In order for the equality [tex]\(5 = 5^x\)[/tex] to hold true, the exponents must be the same. Therefore:
[tex]\[ x = 1 \][/tex]
After following these steps, we find that the value of [tex]\( x \)[/tex] is:
[tex]\[ x = 1 \][/tex]
