Simplify: $7^{\frac{3}{2}} \times 7^{\frac{5}{2}}$
Simplify: $\frac{6^{\frac{11}{2}} \times 6^{-\frac{3}{2}}}{6^4}$
Simplify: $(3^6 \div 3^2) \times 3^{-5}$
Simplify: $(3x^2)^3 \times (4x)^2$
Simplify: $9^2 \times \left(\frac{2}{3}\right)^4$
Simplify: $(5^{-2} + 5^{-1} + 1)^0$
Simplify: $(6^{-1} – 6^{-2}) \div 6^{-3}$
Simplify: $\left(\frac{3}{4}\right)^{-2} \times \left(\frac{5}{3}\right)^{-1} \times \left(\frac{2}{5}\right)^{-2}$
Simplify: $(-1)^{-6} \times (-1)^9 \times (-1)^{-3}$
Simplify: $(-3)^{-8} \div (-3)^{-4}$
Find $x$ if: $\left(\frac{3}{5}\right)^{-2x} = \left(\frac{5}{3}\right)^4$
Rule: When multiplying like bases, add the exponents ($a^m \times a^n = a^{m+n}$).
Step: $\frac{3}{2} + \frac{5}{2} = \frac{8}{2} = 4$.
Result: $7^4 = 2,401$.
Step 1 (Numerator): Add the exponents: $\frac{11}{2} + (-\frac{3}{2}) = \frac{8}{2} = 4$. So, the top becomes $6^4$.
Step 2 (Division): $\frac{6^4}{6^4} = 1$.
Result: $1$.
Step 1 (Parentheses): Subtract exponents for division ($a^m \div a^n = a^{m-n}$): $6 – 2 = 4$.
Step 2 (Multiply): $3^4 \times 3^{-5} = 3^{4 + (-5)} = 3^{-1}$.
Result: $\frac{1}{3}$.
Step 1 (Distribute Exponents): $(3^3 \cdot x^{2 \times 3}) \times (4^2 \cdot x^2) = (27x^6) \times (16x^2)$.
Step 2 (Combine): Multiply coefficients ($27 \times 16 = 432$) and add exponents for $x$ ($6 + 2 = 8$).
Result: $432x^8$.
Step 1: Rewrite $9$ as $3^2$: $(3^2)^2 \times \frac{2^4}{3^4} = 3^4 \times \frac{2^4}{3^4}$.
Step 2: The $3^4$ terms cancel each other out.
Result: $2^4 = 16$.
Rule: Any non-zero expression raised to the power of $0$ is $1$.
Result: $1$.
Step 1 (Expand): $(\frac{1}{6} – \frac{1}{36}) \div \frac{1}{216}$.
Step 2 (Subtraction): Common denominator is $36 \rightarrow (\frac{6}{36} – \frac{1}{36}) = \frac{5}{36}$.
Step 3 (Divide): $\frac{5}{36} \times 216 = 5 \times 6 = 30$.
Result: $30$.
Step 1 (Flip Fractions): Negative exponents flip the base: $(\frac{4}{3})^2 \times (\frac{3}{5})^1 \times (\frac{5}{2})^2$.
Step 2 (Square): $(\frac{16}{9}) \times (\frac{3}{5}) \times (\frac{25}{4})$.
Step 3 (Cancel/Multiply): $(\frac{16}{4}) \times (\frac{25}{5}) \times (\frac{3}{9}) = 4 \times 5 \times \frac{1}{3} = \frac{20}{3}$.
Result: $\frac{20}{3}$ (Note: Correction from previous key, simplified result is $\frac{20}{3}$).
Step: Add all exponents: $-6 + 9 + (-3) = 0$.
Step: $(-1)^0 = 1$.
Result: $1$.
Step: Subtract exponents: $-8 – (-4) = -8 + 4 = -4$.
Result: $(-3)^{-4}$ or $\frac{1}{(-3)^4} = \frac{1}{81}$.
Step 1 (Match Bases): Flip the left side to $(\frac{5}{3})^{2x}$ or the right side to $(\frac{3}{5})^{-4}$. Let’s flip the right: $(\frac{3}{5})^{-2x} = (\frac{3}{5})^{-4}$.
Step 2 (Equalize Exponents): Since bases match, $-2x = -4$.
Result: $x = 2$.