Evaluate (3x^-3y^2*x^-4y^-4)/(3xy^2*3yx^-1)

Solution:

Step 1: Combine the exponents of x terms.

• Add the exponents of $x^{-3}$ and $x^{-4}$.

Step 1.1

• Rewrite the expression as $\frac{3(x^{-4}{x}^{-3})y^2{y}^{-4}}{3x{y}^2(3y{x}^{-1})}$.

Step 1.2

• Apply the exponent rule $a^m\cdot a^n=a^{m+n}$ to get $\frac{3x^{-4-3}{y}^2{y}^{-4}}{3x{y}^2(3y{x}^{-1})}$.

Step 1.3

• Calculate $-4-3$ to simplify the exponent of x to $-7$, resulting in $\frac{3x^{-7}{y}^2{y}^{-4}}{3x{y}^2(3y{x}^{-1})}$.

Step 2: Combine the exponents of y terms in the numerator.

• Add the exponents of $y^2$ and $y^{-4}$.

Step 2.1

• Rewrite the expression as $\frac{3x^{-7}(y^{-4}{y}^2)}{3x{y}^2(3y{x}^{-1})}$.

Step 2.2

• Apply the exponent rule to get $\frac{3x^{-7}{y}^{-4+2}}{3x{y}^2(3y{x}^{-1})}$.

Step 2.3

• Calculate $-4+2$ to simplify the exponent of y to $-2$, resulting in $\frac{3x^{-7}{y}^{-2}}{3x{y}^2(3y{x}^{-1})}$.

Step 3: Combine the exponents of x terms in the denominator.

• Add the exponents of $x^{-1}$ and $x$.

Step 3.1

• Rewrite the expression as $\frac{3x^{-7}{y}^{-2}}{3(x^{-1}x)y^2(3y)}$.

Step 3.2

• Apply the exponent rule to get $\frac{3x^{-7}{y}^{-2}}{3x^{-1+1}{y}^2(3y)}$.

Step 3.3

• Calculate $-1+1$ to simplify the exponent of x to $0$, resulting in $\frac{3x^{-7}{y}^{-2}}{3x^0{y}^2(3y)}$.

Step 4: Simplify the expression with $x^0$.

• Since $x^0=1$, simplify the denominator to $3y^2(3y)$.

Step 5: Combine the exponents of y terms in the denominator.

• Add the exponents of $y$ and $y^2$.

Step 5.1

• Rewrite the expression as $\frac{3x^{-7}{y}^{-2}}{3(y\cdot y^2)\cdot 3}$.

Step 5.2

• Apply the exponent rule to get $\frac{3x^{-7}{y}^{-2}}{3y^{1+2}\cdot 3}$.

Step 5.3

• Calculate $1+2$ to simplify the exponent of y to $3$, resulting in $\frac{3x^{-7}{y}^{-2}}{3y^3\cdot 3}$.

Step 6: Move $x^{-7}$ to the denominator using the negative exponent rule.

• Apply the rule $b^{-n}=\frac{1}{b^n}$ to get $\frac{3y^{-2}}{3y^3\cdot 3x^7}$.

Step 7: Move $y^{-2}$ to the denominator using the negative exponent rule.

• Apply the rule to get $\frac{3}{3y^3\cdot 3x^7{y}^2}$.

Step 8: Combine the exponents of y terms in the denominator.

• Add the exponents of $y^3$ and $y^2$.

Step 8.1

• Rewrite the expression as $\frac{3}{3(y^2{y}^3)\cdot 3x^7}$.

Step 8.2

• Apply the exponent rule to get $\frac{3}{3y^{2+3}\cdot 3x^7}$.

Step 8.3

• Calculate $2+3$ to simplify the exponent of y to $5$, resulting in $\frac{3}{3y^5\cdot 3x^7}$.

Step 9: Cancel the common factor of 3.

• Simplify the expression by canceling the common factor to get $\frac{1}{y^5\cdot 3x^7}$.

Step 10: Rearrange the terms in the denominator.

• Place the constant 3 to the left of $y^5$ to get the final result $\frac{1}{3y^5{x}^7}$.

Knowledge Notes:

The solution involves several key algebraic rules for manipulating exponents:

1. Power Rule: $a^m\cdot a^n=a^{m+n}$. This rule is used to combine like bases with exponents by adding the exponents together.

2. Negative Exponent Rule: $b^{-n}=\frac{1}{b^n}$. This rule is used to transform a term with a negative exponent into a reciprocal with a positive exponent.

3. Zero Exponent Rule: $x^0=1$. Any nonzero number raised to the power of zero is equal to one.

4. Combining Like Terms: When simplifying expressions, like terms can be combined by adding or subtracting their coefficients.

5. Simplification: Common factors in the numerator and denominator can be canceled out to simplify the expression.

The problem-solving process involves applying these rules step by step to simplify the given algebraic fraction. The steps are carefully sequenced to address one operation at a time, ensuring that the simplification process is clear and methodical.