1. Simplify $\frac{2}{3} \times \frac{2}{4}$. Use the rule for multiplying fractions: multiply numerators and denominators. $$\frac{2}{3} \times \frac{2}{4} = \frac{2 \times 2}{3 \times 4} = \frac{4}{12}$$ Simplify $\frac{4}{12}$ by dividing numerator and denominator by 4: $$\frac{4}{12} = \frac{1}{3}$$ 2. Simplify $x^{5} \times x^{2}$. Use the rule for multiplying powers with the same base: add exponents. $$x^{5} \times x^{2} = x^{5+2} = x^{7}$$ 3. Simplify $5^{1} \times 5^{6}$. Add exponents: $$5^{1} \times 5^{6} = 5^{1+6} = 5^{7}$$ 4. Simplify $a^{7} \times a^{3}$. Add exponents: $$a^{7} \times a^{3} = a^{7+3} = a^{10}$$ 5. Simplify $3^{4} \times 3^{5}$. Add exponents: $$3^{4} \times 3^{5} = 3^{4+5} = 3^{9}$$ 6. Simplify $y^{2} \times y^{8}$. Add exponents: $$y^{2} \times y^{8} = y^{2+8} = y^{10}$$ 7. Simplify $7^{0} \times 7^{9}$. Recall $7^{0} = 1$. $$7^{0} \times 7^{9} = 1 \times 7^{9} = 7^{9}$$ 8. Simplify $b^{10} \times b^{0}$. Recall $b^{0} = 1$. $$b^{10} \times b^{0} = b^{10} \times 1 = b^{10}$$ 9. Simplify $4^{3} \times 4^{2}$. Add exponents: $$4^{3} \times 4^{2} = 4^{3+2} = 4^{5}$$ 10. Simplify $z^{6} \times z^{1}$. Add exponents: $$z^{6} \times z^{1} = z^{6+1} = z^{7}$$