Solve sodda 1 ) (a ^ 3 + 6b ^ 2) ^ 2 - (6b ^ 2 - a ^ 3) ^ 2 | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

$24b^{2}a^{3}$

Solution Steps

Use binomial theorem $\left(p+q\right)^{2}=p^{2}+2pq+q^{2}$ to expand $\left(a^{3}+6b^{2}\right)^{2}$.

$$\left(a^{3}\right)^{2}+12a^{3}b^{2}+36\left(b^{2}\right)^{2}-\left(6b^{2}-a^{3}\right)^{2}$$

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

$$a^{6}+12a^{3}b^{2}+36\left(b^{2}\right)^{2}-\left(6b^{2}-a^{3}\right)^{2}$$

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

$$a^{6}+12a^{3}b^{2}+36b^{4}-\left(6b^{2}-a^{3}\right)^{2}$$

Use binomial theorem $\left(p-q\right)^{2}=p^{2}-2pq+q^{2}$ to expand $\left(6b^{2}-a^{3}\right)^{2}$.

$$a^{6}+12a^{3}b^{2}+36b^{4}-\left(36\left(b^{2}\right)^{2}-12b^{2}a^{3}+\left(a^{3}\right)^{2}\right)$$

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

$$a^{6}+12a^{3}b^{2}+36b^{4}-\left(36b^{4}-12b^{2}a^{3}+\left(a^{3}\right)^{2}\right)$$

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

$$a^{6}+12a^{3}b^{2}+36b^{4}-\left(36b^{4}-12b^{2}a^{3}+a^{6}\right)$$

To find the opposite of $36b^{4}-12b^{2}a^{3}+a^{6}$, find the opposite of each term.

$$a^{6}+12a^{3}b^{2}+36b^{4}-36b^{4}+12b^{2}a^{3}-a^{6}$$

Combine $36b^{4}$ and $-36b^{4}$ to get $0$.

$$a^{6}+12a^{3}b^{2}+12b^{2}a^{3}-a^{6}$$

Combine $12a^{3}b^{2}$ and $12b^{2}a^{3}$ to get $24a^{3}b^{2}$.

$$a^{6}+24a^{3}b^{2}-a^{6}$$

Combine $a^{6}$ and $-a^{6}$ to get $0$.

$$24a^{3}b^{2}$$

Expand

$24b^{2}a^{3}$

Solution Steps

Use binomial theorem $\left(p+q\right)^{2}=p^{2}+2pq+q^{2}$ to expand $\left(a^{3}+6b^{2}\right)^{2}$.

$$\left(a^{3}\right)^{2}+12a^{3}b^{2}+36\left(b^{2}\right)^{2}-\left(6b^{2}-a^{3}\right)^{2}$$

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

$$a^{6}+12a^{3}b^{2}+36\left(b^{2}\right)^{2}-\left(6b^{2}-a^{3}\right)^{2}$$

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

$$a^{6}+12a^{3}b^{2}+36b^{4}-\left(6b^{2}-a^{3}\right)^{2}$$

Use binomial theorem $\left(p-q\right)^{2}=p^{2}-2pq+q^{2}$ to expand $\left(6b^{2}-a^{3}\right)^{2}$.

$$a^{6}+12a^{3}b^{2}+36b^{4}-\left(36\left(b^{2}\right)^{2}-12b^{2}a^{3}+\left(a^{3}\right)^{2}\right)$$

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

$$a^{6}+12a^{3}b^{2}+36b^{4}-\left(36b^{4}-12b^{2}a^{3}+\left(a^{3}\right)^{2}\right)$$

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

$$a^{6}+12a^{3}b^{2}+36b^{4}-\left(36b^{4}-12b^{2}a^{3}+a^{6}\right)$$

To find the opposite of $36b^{4}-12b^{2}a^{3}+a^{6}$, find the opposite of each term.

$$a^{6}+12a^{3}b^{2}+36b^{4}-36b^{4}+12b^{2}a^{3}-a^{6}$$

Combine $36b^{4}$ and $-36b^{4}$ to get $0$.

$$a^{6}+12a^{3}b^{2}+12b^{2}a^{3}-a^{6}$$

Combine $12a^{3}b^{2}$ and $12b^{2}a^{3}$ to get $24a^{3}b^{2}$.

$$a^{6}+24a^{3}b^{2}-a^{6}$$

Combine $a^{6}$ and $-a^{6}$ to get $0$.

$$24a^{3}b^{2}$$