Solve (-10t+30t^2)^3

$$( - 10 t + 30 t ^ { 2 } ) ^ { 3 }$$

Evaluate

$$1000\left(t\left(3t-1\right)\right)^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(-10t+30t^{2}\right)^{3}.

$$-1000t^{3}+9000t^{2}t^{2}-27000t\left(t^{2}\right)^{2}+27000\left(t^{2}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$-1000t^{3}+9000t^{4}-27000t\left(t^{2}\right)^{2}+27000\left(t^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$-1000t^{3}+9000t^{4}-27000tt^{4}+27000\left(t^{2}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$-1000t^{3}+9000t^{4}-27000t^{5}+27000\left(t^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$-1000t^{3}+9000t^{4}-27000t^{5}+27000t^{6}$$

Expand

$$27000t^{6}-27000t^{5}+9000t^{4}-1000t^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(-10t+30t^{2}\right)^{3}.

$$-1000t^{3}+9000t^{2}t^{2}-27000t\left(t^{2}\right)^{2}+27000\left(t^{2}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$-1000t^{3}+9000t^{4}-27000t\left(t^{2}\right)^{2}+27000\left(t^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$-1000t^{3}+9000t^{4}-27000tt^{4}+27000\left(t^{2}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$-1000t^{3}+9000t^{4}-27000t^{5}+27000\left(t^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$-1000t^{3}+9000t^{4}-27000t^{5}+27000t^{6}$$