Simplify the following indices: a) 18a2b/6ab b) y5xy2 c) 6^-3 d) a2b^-2 / a3b^-2

a) Simplify $18a^3b^2 \div 6ab$ (Assuming the '=' sign in the question is a typo and should be '$\div$' for simplification.)

Step 1: Rewrite the expression as a fraction. $\frac{18a^3b^2}{6ab}$ Step 2: Divide the numerical coefficients. $\frac{18}{6} = 3$ Step 3: Apply the quotient rule for exponents ($x^m \div x^n = x^{m-n}$) for variables $a$ and $b$. For $a$: $a^{3-1} = a^2$ For $b$: $b^{2-1} = b^1 = b$ Step 4: Combine the simplified parts. $3a^2b$ The simplified expression is $\boxed{3a^2b}$.

b) Simplify $y^3 \times y^2$

Step 1: Apply the product rule for exponents ($x^m \times x^n = x^{m+n}$). $y^{3+2}$ Step 2: Add the exponents. $y^5$ The simplified expression is $\boxed{y^5}$.

c) Simplify $6^{-3}$

Step 1: Apply the negative exponent rule ($x^{-n} = \frac{1}{x^n}$). $\frac{1}{6^3}$ Step 2: Calculate the value of $6^3$. $6^3 = 6 \times 6 \times 6 = 216$ Step 3: Substitute the value back into the expression. $\frac{1}{216}$ The simplified expression is \boxed{\frac{1{216}}}.

d) Simplify $\frac{a^2b^{-2}}{a^3b^{-2}}$

Step 1: Apply the quotient rule for exponents for variable $a$. $a^{2-3} = a^{-1}$ Step 2: Apply the quotient rule for exponents for variable $b$. $b^{-2 - (-2)} = b^{-2+2} = b^0$ Step 3: Apply the zero exponent rule ($x^0 = 1$). $b^0 = 1$ Step 4: Combine the simplified parts. $a^{-1} \times 1 = a^{-1}$ Step 5: Apply the negative exponent rule to express $a^{-1}$ with a positive exponent. $\frac{1}{a}$ The simplified expression is \boxed{\frac{1{a}}}.

e) Simplify $18a^3b^2 \div 3ab$

Step 1: Rewrite the expression as a fraction. $\frac{18a^3b^2}{3ab}$ Step 2: Divide the numerical coefficients. $\frac{18}{3} = 6$ Step 3: Apply the quotient rule for exponents ($x^m \div x^n = x^{m-n}$) for variables $a$ and $b$. For $a$: $a^{3-1} = a^2$ For $b$: $b^{2-1} = b^1 = b$ Step 4: Combine the simplified parts. $6a^2b$ The simplified expression is $\boxed{6a^2b}$.

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