Simplify (6y)/((7x^2)/((9y^2)/(14x^4)))

Solution:

Step 1:

Start by inverting the complex fraction's denominator and multiplying it with the numerator. $\frac{6y}{\frac{7x^2}{\frac{9y^2}{14x^4}}}$

Step 2:

Rewrite the expression to show multiplication by the reciprocal. $\frac{6y}{1}\cdot \frac{14x^4}{7x^2}\cdot \frac{1}{9y^2}$

Step 3:

Simplify the expression by combining the fractions. $\frac{6y\cdot 14x^4}{7x^2\cdot 9y^2}$

Step 4:

Simplify the powers of $x$ by adding the exponents.

Step 4.1:

Rearrange the terms to prepare for exponent addition. $\frac{6y\cdot 14}{7\cdot 9y^2}\cdot x^{4+2}$

Step 4.2:

Apply the power rule for exponents, $a^m\cdot a^n=a^{m+n}$. $\frac{6y\cdot 14}{7\cdot 9y^2}\cdot x^6$

Step 4.3:

Calculate the sum of the exponents for $x$. $\frac{6y\cdot 14}{63y^2}\cdot x^6$

Step 5:

Perform the multiplication of the constants and variables.

Step 5.1:

Multiply the numerator's constants. $\frac{6y\cdot 14}{63y^2}\cdot x^6$

Step 5.2:

Simplify the fraction by multiplying the constants in the denominator. $\frac{84y}{63y^2}\cdot x^6$

Step 6:

Reduce the fraction by eliminating common factors.

Step 6.1:

Extract the common factor from the numerator. $2\cdot \frac{3y}{63y^2}\cdot x^6$

Step 6.2:

Extract the common factor from the denominator. $2\cdot \frac{3y}{2\cdot 31.5y^2}\cdot x^6$

Step 6.3:

Cancel out the common factors. $\frac{3y}{31.5y^2}\cdot x^6$

Step 6.4:

Rewrite the simplified expression. $\frac{3y}{31.5y^2}\cdot x^6$

Step 7:

Combine the constants and variables in the numerator. $\frac{3\cdot 9y^3}{31.5\cdot x^6}$

Step 8:

Multiply the constants in the numerator. $\frac{27y^3}{31.5x^6}$

Step 9:

Combine the variable $y$ with the rest of the numerator. $\frac{27y^3}{31.5x^6}$

Step 10:

Express $y$ with an exponent. $\frac{27y^{1+2}}{31.5x^6}$

Step 11:

Use the power rule for exponents to combine $y$ terms. $\frac{27y^3}{31.5x^6}$

Step 12:

Add the exponents for $y$. $\frac{27y^3}{31.5x^6}$

Knowledge Notes:

1. Reciprocal Multiplication: When dividing by a fraction, you can multiply by its reciprocal (i.e., flip the numerator and denominator).

2. Combining Fractions: To combine fractions, you can multiply the numerators and denominators respectively.

3. Exponent Addition: The power rule for exponents states that when multiplying like bases, you add the exponents: $a^m\cdot a^n=a^{m+n}$.

4. Simplifying Expressions: Simplify expressions by canceling out common factors in the numerator and denominator.

5. Multiplying Constants and Variables: When multiplying constants and variables, multiply the coefficients (numbers) and add the exponents if the bases are the same.

6. Reducing Fractions: To reduce fractions, divide both the numerator and the denominator by their greatest common factor.