Answer:
7. [tex]\sqrt[3]{\frac{7}{3}}[/tex]
8. [tex]\dfrac{\sqrt[3]{7} \cdot \sqrt{3}}{3}[/tex]
9. [tex]\sqrt{5} + \sqrt{2}[/tex]
Step-by-step explanation:
Law of Indices:
The laws of indices or exponents are rules that govern the manipulation of expressions involving powers.
Here are some fundamental laws:
1. Product Rule: [tex]a^m \cdot a^n = a^{m + n}[/tex]
2. Quotient Rule: [tex]\frac{a^m}{a^n} = a^{m - n}[/tex]
3. Power Rule: [tex](a^m)^n = a^{mn}[/tex]
4. Root Rule: [tex]\sqrt[n]{a^m} = a^{\frac{m}{n}}[/tex]
Now, let's simplify the given expressions using these rules:
7. [tex]\sqrt[3]{7} \div \sqrt[3]{3}[/tex]:
Using the quotient rule for roots:
[tex] \sqrt[3]{7} \div \sqrt[3]{3} = \sqrt[3]{\dfrac{7}{3}} [/tex]
8. [tex]\sqrt[3]{7} \div \sqrt{3}[/tex]:
Using the quotient rule for roots:
[tex] \sqrt[3]{7} \div \sqrt{3} = \sqrt[3]{\dfrac{7}{3}} \cdot \dfrac{1}{\sqrt{3}} [/tex]
Rationalize the denominator by multiplying the numerator and denominator by [tex]\sqrt{3}[/tex]:
[tex] = \dfrac{\sqrt[3]{7} \cdot \sqrt{3}}{3} [/tex]
9. [tex]3 \div (\sqrt{5} - \sqrt{2})[/tex]:
To simplify this expression, we'll use the conjugate to eliminate the square root in the denominator.
Multiply the numerator and denominator by the conjugate of the denominator:
[tex] \dfrac{3 \cdot (\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2}) \cdot (\sqrt{5} + \sqrt{2})} [/tex]
Multiply the numerators and denominators:
[tex] = \dfrac{3(\sqrt{5} + \sqrt{2})}{5 - 2} [/tex]
[tex] = \dfrac{3(\sqrt{5} + \sqrt{2})}{3} [/tex]
[tex] = \sqrt{5} + \sqrt{2} [/tex]
So, the simplified expressions are:
7. [tex]\sqrt[3]{\frac{7}{3}}[/tex]
8. [tex]\dfrac{\sqrt[3]{7} \cdot \sqrt{3}}{3}[/tex]
9. [tex]\sqrt{5} + \sqrt{2}[/tex]
