Simplify: $3^{-2} \times (-5)^3$
Simplify: $(-6)^7 \div (-6)^3$
Simplify: $\left(\frac{3}{4}\right)^{-4}$
Simplify: $\frac{9^{-2} \times 3^5}{3^{-3}}$
Simplify: $\left(\frac{2}{5}\right)^{-1} – \left(\frac{3}{5}\right)^{-1}$
Simplify: $\left(\frac{7}{9}\right)^{-3} \times \left(\frac{3}{7}\right)^{-2}$
Simplify: $\frac{16x^{-3}}{4^{-1} \times x^2}$
Simplify: $\frac{5^{-2} \times 25^2 \times 125}{5^4 \times 5^{-1}}$
Simplify: $\left\{ \left( \frac{3}{5} \right)^2 \right\}^3 \times \left( \frac{5}{3} \right)^{-4} \times 5^{-1} \times 3^2$
Find $x$, if: $\left( \frac{5}{2} \right)^{x+1} = \left( \frac{5}{2} \right)^4$
If $x = \left( \frac{4}{3} \right)^3 \times \left( \frac{3}{2} \right)^2$, find the valueof $x^2$
1. Simplify: $3^{-2} \times (-5)^3$
Step: Convert the negative exponent to a fraction: $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.
Step: Calculate the cube: $(-5) \times (-5) \times (-5) = -125$.
Final: $\frac{1}{9} \times -125 = \mathbf{-\frac{125}{9}}$.
2. Simplify: $(-6)^7 \div (-6)^3$
Rule: When dividing same bases, subtract exponents ($a^m \div a^n = a^{m-n}$).
Step: $(-6)^{7-3} = (-6)^4$.
Final: $(-6) \times (-6) \times (-6) \times (-6) = \mathbf{1,296}$.
3. Simplify: $\left(\frac{3}{4}\right)^{-4}$
Rule: A negative exponent on a fraction flips the fraction: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.
Step: $\left(\frac{4}{3}\right)^4 = \frac{4^4}{3^4}$.
Final: $\frac{256}{81}$.
4. Simplify: $\frac{9^{-2} \times 3^5}{3^{-3}}$
Step: Change base 9 to base 3: $9 = 3^2$, so $9^{-2} = (3^2)^{-2} = 3^{-4}$.
Step: Combine the top: $3^{-4} \times 3^5 = 3^{-4+5} = 3^1$.
Step: Divide: $\frac{3^1}{3^{-3}} = 3^{1 – (-3)} = 3^4$.
Final: $\mathbf{81}$.
5. Simplify: $\left(\frac{2}{5}\right)^{-1} – \left(\frac{3}{5}\right)^{-1}$
Step: Apply the negative exponent: $\frac{5}{2} – \frac{5}{3}$.
Step: Find a common denominator (6): $\frac{15}{6} – \frac{10}{6}$.
Final: $\mathbf{\frac{5}{6}}$.
6. Simplify: $\left(\frac{7}{9}\right)^{-3} \times \left(\frac{3}{7}\right)^{-2}$
Step: Flip fractions: $\left(\frac{9}{7}\right)^3 \times \left(\frac{7}{3}\right)^2 = \frac{9^3}{7^3} \times \frac{7^2}{3^2}$.
Step: Simplify base 9: $\frac{(3^2)^3 \cdot 7^2}{7^3 \cdot 3^2} = \frac{3^6 \cdot 7^2}{7^3 \cdot 3^2}$.
Final: Subtract exponents: $3^{6-2} \cdot 7^{2-3} = 3^4 \cdot 7^{-1} = \mathbf{\frac{81}{7}}$.
7. Simplify: $\frac{16x^{-3}}{4^{-1} \times x^2}$
Step: Move negative exponents across the fraction bar: $\frac{16 \cdot 4^1}{x^2 \cdot x^3}$.
Final: $\mathbf{\frac{64}{x^5}}$.
8. Simplify: $\frac{5^{-2} \times 25^2 \times 125}{5^4 \times 5^{-1}}$
Step: Convert to base 5: $25^2 = (5^2)^2 = 5^4$ and $125 = 5^3$.
Step: Simplify numerator: $5^{-2} \times 5^4 \times 5^3 = 5^{-2+4+3} = 5^5$.
Step: Simplify denominator: $5^4 \times 5^{-1} = 5^3$.
Final: $5^5 \div 5^3 = 5^{5-3} = 5^2 = \mathbf{25}$.
9. Simplify: $\left\{ \left( \frac{3}{5} \right)^2 \right\}^3 \times \left( \frac{5}{3} \right)^{-4} \times 5^{-1} \times 3^2$
Step: Apply power of a power: $(\frac{3}{5})^{2 \times 3} = (\frac{3}{5})^6$.
Step: Flip the negative exponent: $(\frac{5}{3})^{-4} = (\frac{3}{5})^4$.
Step: Group bases: $\frac{3^6}{5^6} \times \frac{3^4}{5^4} \times \frac{1}{5} \times 3^2 = \frac{3^{6+4+2}}{5^{6+4+1}} = \frac{3^{12}}{5^{11}}$.
10. Find $x$, if: $\left( \frac{5}{2} \right)^{x+1} = \left( \frac{5}{2} \right)^4$
Rule: If bases are equal, the exponents must be equal.
Step: $x + 1 = 4$.
Final: $x = 4 – 1 \Rightarrow \mathbf{x = 3}$.
11. If $x = \left( \frac{4}{3} \right)^3 \times \left( \frac{3}{2} \right)^2$, find the value of $x^2$.
Step: Simplify $x$: $\frac{4^3}{3^3} \times \frac{3^2}{2^2} = \frac{(2^2)^3 \cdot 3^2}{3^3 \cdot 2^2} = \frac{2^6 \cdot 3^2}{3^3 \cdot 2^2}$.
Step: $x = 2^{6-2} \cdot 3^{2-3} = 2^4 \cdot 3^{-1} = \frac{16}{3}$.
Final: Find $x^2$: $(\frac{16}{3})^2 = \mathbf{\frac{256}{81}}$.