Solve ((3)/(2)b+(2)/(3b))((9(b)^(2))/(4)+(4)/(9(b)^(2))-1) | 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

$$( \frac{ 3 }{ 2 } b+ \frac{ 2 }{ 3b } )( \frac{ 9 { b }^{ 2 } }{ 4 } + \frac{ 4 }{ 9 { b }^{ 2 } } -1)$$

Evaluate

$\frac{27b^{3}}{8}+\frac{8}{27b^{3}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4$ and $9b^{2}$ is $36b^{2}$. Multiply $\frac{9b^{2}}{4}$ times $\frac{9b^{2}}{9b^{2}}$. Multiply $\frac{4}{9b^{2}}$ times $\frac{4}{4}$.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\left(\frac{9b^{2}\times 9b^{2}}{36b^{2}}+\frac{4\times 4}{36b^{2}}-1\right)$$

Since $\frac{9b^{2}\times 9b^{2}}{36b^{2}}$ and $\frac{4\times 4}{36b^{2}}$ have the same denominator, add them by adding their numerators.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\left(\frac{9b^{2}\times 9b^{2}+4\times 4}{36b^{2}}-1\right)$$

Do the multiplications in $9b^{2}\times 9b^{2}+4\times 4$.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\left(\frac{81b^{4}+16}{36b^{2}}-1\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $1$ times $\frac{36b^{2}}{36b^{2}}$.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\left(\frac{81b^{4}+16}{36b^{2}}-\frac{36b^{2}}{36b^{2}}\right)$$

Since $\frac{81b^{4}+16}{36b^{2}}$ and $\frac{36b^{2}}{36b^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}$$

Use the distributive property to multiply $\frac{3}{2}b+\frac{2}{3b}$ by $\frac{81b^{4}+16-36b^{2}}{36b^{2}}$.

$$\frac{3}{2}b\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}+\frac{2}{3b}\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}$$

Multiply $\frac{3}{2}$ times $\frac{81b^{4}+16-36b^{2}}{36b^{2}}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(81b^{4}+16-36b^{2}\right)}{2\times 36b^{2}}b+\frac{2}{3b}\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{81b^{4}-36b^{2}+16}{2\times 12b^{2}}b+\frac{2}{3b}\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}$$

Multiply $\frac{2}{3b}$ times $\frac{81b^{4}+16-36b^{2}}{36b^{2}}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{81b^{4}-36b^{2}+16}{2\times 12b^{2}}b+\frac{2\left(81b^{4}+16-36b^{2}\right)}{3b\times 36b^{2}}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{81b^{4}-36b^{2}+16}{2\times 12b^{2}}b+\frac{81b^{4}-36b^{2}+16}{3\times 18bb^{2}}$$

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

$$\frac{81b^{4}-36b^{2}+16}{2\times 12b^{2}}b+\frac{81b^{4}-36b^{2}+16}{3\times 18b^{3}}$$

Multiply $2$ and $12$ to get $24$.

$$\frac{81b^{4}-36b^{2}+16}{24b^{2}}b+\frac{81b^{4}-36b^{2}+16}{3\times 18b^{3}}$$

Express $\frac{81b^{4}-36b^{2}+16}{24b^{2}}b$ as a single fraction.

$$\frac{\left(81b^{4}-36b^{2}+16\right)b}{24b^{2}}+\frac{81b^{4}-36b^{2}+16}{3\times 18b^{3}}$$

Cancel out $b$ in both numerator and denominator.

$$\frac{81b^{4}-36b^{2}+16}{24b}+\frac{81b^{4}-36b^{2}+16}{3\times 18b^{3}}$$

Multiply $3$ and $18$ to get $54$.

$$\frac{81b^{4}-36b^{2}+16}{24b}+\frac{81b^{4}-36b^{2}+16}{54b^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $24b$ and $54b^{3}$ is $216b^{3}$. Multiply $\frac{81b^{4}-36b^{2}+16}{24b}$ times $\frac{9b^{2}}{9b^{2}}$. Multiply $\frac{81b^{4}-36b^{2}+16}{54b^{3}}$ times $\frac{4}{4}$.

$$\frac{\left(81b^{4}-36b^{2}+16\right)\times 9b^{2}}{216b^{3}}+\frac{4\left(81b^{4}-36b^{2}+16\right)}{216b^{3}}$$

Since $\frac{\left(81b^{4}-36b^{2}+16\right)\times 9b^{2}}{216b^{3}}$ and $\frac{4\left(81b^{4}-36b^{2}+16\right)}{216b^{3}}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(81b^{4}-36b^{2}+16\right)\times 9b^{2}+4\left(81b^{4}-36b^{2}+16\right)}{216b^{3}}$$

Do the multiplications in $\left(81b^{4}-36b^{2}+16\right)\times 9b^{2}+4\left(81b^{4}-36b^{2}+16\right)$.

$$\frac{729b^{6}-324b^{4}+144b^{2}+324b^{4}-144b^{2}+64}{216b^{3}}$$

Combine like terms in $729b^{6}-324b^{4}+144b^{2}+324b^{4}-144b^{2}+64$.

$$\frac{729b^{6}+64}{216b^{3}}$$

Expand

$\frac{27b^{3}}{8}+\frac{8}{27b^{3}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4$ and $9b^{2}$ is $36b^{2}$. Multiply $\frac{9b^{2}}{4}$ times $\frac{9b^{2}}{9b^{2}}$. Multiply $\frac{4}{9b^{2}}$ times $\frac{4}{4}$.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\left(\frac{9b^{2}\times 9b^{2}}{36b^{2}}+\frac{4\times 4}{36b^{2}}-1\right)$$

Since $\frac{9b^{2}\times 9b^{2}}{36b^{2}}$ and $\frac{4\times 4}{36b^{2}}$ have the same denominator, add them by adding their numerators.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\left(\frac{9b^{2}\times 9b^{2}+4\times 4}{36b^{2}}-1\right)$$

Do the multiplications in $9b^{2}\times 9b^{2}+4\times 4$.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\left(\frac{81b^{4}+16}{36b^{2}}-1\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $1$ times $\frac{36b^{2}}{36b^{2}}$.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\left(\frac{81b^{4}+16}{36b^{2}}-\frac{36b^{2}}{36b^{2}}\right)$$

Since $\frac{81b^{4}+16}{36b^{2}}$ and $\frac{36b^{2}}{36b^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{3}{2}b+\frac{2}{3b}\right)\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}$$

Use the distributive property to multiply $\frac{3}{2}b+\frac{2}{3b}$ by $\frac{81b^{4}+16-36b^{2}}{36b^{2}}$.

$$\frac{3}{2}b\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}+\frac{2}{3b}\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}$$

Multiply $\frac{3}{2}$ times $\frac{81b^{4}+16-36b^{2}}{36b^{2}}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(81b^{4}+16-36b^{2}\right)}{2\times 36b^{2}}b+\frac{2}{3b}\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{81b^{4}-36b^{2}+16}{2\times 12b^{2}}b+\frac{2}{3b}\times \frac{81b^{4}+16-36b^{2}}{36b^{2}}$$

Multiply $\frac{2}{3b}$ times $\frac{81b^{4}+16-36b^{2}}{36b^{2}}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{81b^{4}-36b^{2}+16}{2\times 12b^{2}}b+\frac{2\left(81b^{4}+16-36b^{2}\right)}{3b\times 36b^{2}}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{81b^{4}-36b^{2}+16}{2\times 12b^{2}}b+\frac{81b^{4}-36b^{2}+16}{3\times 18bb^{2}}$$

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

$$\frac{81b^{4}-36b^{2}+16}{2\times 12b^{2}}b+\frac{81b^{4}-36b^{2}+16}{3\times 18b^{3}}$$

Multiply $2$ and $12$ to get $24$.

$$\frac{81b^{4}-36b^{2}+16}{24b^{2}}b+\frac{81b^{4}-36b^{2}+16}{3\times 18b^{3}}$$

Express $\frac{81b^{4}-36b^{2}+16}{24b^{2}}b$ as a single fraction.

$$\frac{\left(81b^{4}-36b^{2}+16\right)b}{24b^{2}}+\frac{81b^{4}-36b^{2}+16}{3\times 18b^{3}}$$

Cancel out $b$ in both numerator and denominator.

$$\frac{81b^{4}-36b^{2}+16}{24b}+\frac{81b^{4}-36b^{2}+16}{3\times 18b^{3}}$$

Multiply $3$ and $18$ to get $54$.

$$\frac{81b^{4}-36b^{2}+16}{24b}+\frac{81b^{4}-36b^{2}+16}{54b^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $24b$ and $54b^{3}$ is $216b^{3}$. Multiply $\frac{81b^{4}-36b^{2}+16}{24b}$ times $\frac{9b^{2}}{9b^{2}}$. Multiply $\frac{81b^{4}-36b^{2}+16}{54b^{3}}$ times $\frac{4}{4}$.

$$\frac{\left(81b^{4}-36b^{2}+16\right)\times 9b^{2}}{216b^{3}}+\frac{4\left(81b^{4}-36b^{2}+16\right)}{216b^{3}}$$

Since $\frac{\left(81b^{4}-36b^{2}+16\right)\times 9b^{2}}{216b^{3}}$ and $\frac{4\left(81b^{4}-36b^{2}+16\right)}{216b^{3}}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(81b^{4}-36b^{2}+16\right)\times 9b^{2}+4\left(81b^{4}-36b^{2}+16\right)}{216b^{3}}$$

Do the multiplications in $\left(81b^{4}-36b^{2}+16\right)\times 9b^{2}+4\left(81b^{4}-36b^{2}+16\right)$.

$$\frac{729b^{6}-324b^{4}+144b^{2}+324b^{4}-144b^{2}+64}{216b^{3}}$$

Combine like terms in $729b^{6}-324b^{4}+144b^{2}+324b^{4}-144b^{2}+64$.

$$\frac{729b^{6}+64}{216b^{3}}$$