Solve 3b^{3}(-2a)^2+1/2a^2b(-2b)^2-6(-ab)^2(-1/2b)

$$3 b ^ { 3 } ( - 2 a ) ^ { 2 } + \frac { 1 } { 2 } a ^ { 2 } b ( - 2 b ) ^ { 2 } - 6 ( - a b ) ^ { 2 } ( - \frac { 1 } { 2 } b )$$

Evaluate

$$17a^{2}b^{3}$$

Solution Steps

Expand \left(-2a\right)^{2}.

$$3b^{3}\left(-2\right)^{2}a^{2}+\frac{1}{2}a^{2}b\left(-2b\right)^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Calculate -2 to the power of 2 and get 4.

$$3b^{3}\times 4a^{2}+\frac{1}{2}a^{2}b\left(-2b\right)^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Multiply 3 and 4 to get 12.

$$12b^{3}a^{2}+\frac{1}{2}a^{2}b\left(-2b\right)^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Expand \left(-2b\right)^{2}.

$$12b^{3}a^{2}+\frac{1}{2}a^{2}b\left(-2\right)^{2}b^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Calculate -2 to the power of 2 and get 4.

$$12b^{3}a^{2}+\frac{1}{2}a^{2}b\times 4b^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Multiply \frac{1}{2} and 4 to get 2.

$$12b^{3}a^{2}+2a^{2}bb^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$12b^{3}a^{2}+2a^{2}b^{3}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Combine 12b^{3}a^{2} and 2a^{2}b^{3} to get 14b^{3}a^{2}.

$$14b^{3}a^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Expand \left(\left(-a\right)b\right)^{2}.

$$14b^{3}a^{2}-6\left(-a\right)^{2}b^{2}\left(-\frac{1}{2}\right)b$$

Calculate -a to the power of 2 and get a^{2}.

$$14b^{3}a^{2}-6a^{2}b^{2}\left(-\frac{1}{2}\right)b$$

Multiply 6 and -\frac{1}{2} to get -3.

$$14b^{3}a^{2}-\left(-3a^{2}b^{2}b\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$14b^{3}a^{2}-\left(-3a^{2}b^{3}\right)$$

The opposite of -3a^{2}b^{3} is 3a^{2}b^{3}.

$$14b^{3}a^{2}+3a^{2}b^{3}$$

Combine 14b^{3}a^{2} and 3a^{2}b^{3} to get 17b^{3}a^{2}.

$$17b^{3}a^{2}$$

Expand

$$17a^{2}b^{3}$$

Solution Steps

Expand \left(-2a\right)^{2}.

$$3b^{3}\left(-2\right)^{2}a^{2}+\frac{1}{2}a^{2}b\left(-2b\right)^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Calculate -2 to the power of 2 and get 4.

$$3b^{3}\times 4a^{2}+\frac{1}{2}a^{2}b\left(-2b\right)^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Multiply 3 and 4 to get 12.

$$12b^{3}a^{2}+\frac{1}{2}a^{2}b\left(-2b\right)^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Expand \left(-2b\right)^{2}.

$$12b^{3}a^{2}+\frac{1}{2}a^{2}b\left(-2\right)^{2}b^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Calculate -2 to the power of 2 and get 4.

$$12b^{3}a^{2}+\frac{1}{2}a^{2}b\times 4b^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Multiply \frac{1}{2} and 4 to get 2.

$$12b^{3}a^{2}+2a^{2}bb^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$12b^{3}a^{2}+2a^{2}b^{3}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Combine 12b^{3}a^{2} and 2a^{2}b^{3} to get 14b^{3}a^{2}.

$$14b^{3}a^{2}-6\left(\left(-a\right)b\right)^{2}\left(-\frac{1}{2}\right)b$$

Expand \left(\left(-a\right)b\right)^{2}.

$$14b^{3}a^{2}-6\left(-a\right)^{2}b^{2}\left(-\frac{1}{2}\right)b$$

Calculate -a to the power of 2 and get a^{2}.

$$14b^{3}a^{2}-6a^{2}b^{2}\left(-\frac{1}{2}\right)b$$

Multiply 6 and -\frac{1}{2} to get -3.

$$14b^{3}a^{2}-\left(-3a^{2}b^{2}b\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$14b^{3}a^{2}-\left(-3a^{2}b^{3}\right)$$

The opposite of -3a^{2}b^{3} is 3a^{2}b^{3}.

$$14b^{3}a^{2}+3a^{2}b^{3}$$

Combine 14b^{3}a^{2} and 3a^{2}b^{3} to get 17b^{3}a^{2}.

$$17b^{3}a^{2}$$