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How to Simplify the Rational Expression $\frac{y^4}{-2x^2y^4} \cdot -xy$
Mathematics
Grade 8 (Junior High School)
Question Content
Simplify the rational expression: $\frac{y^4}{-2x^2y^4} \cdot -xy$
Correct Answer
$\frac{1}{2x}$
Detailed Solution Steps
1
Step 1: Cancel out the common factor $y^4$ in the numerator and denominator of the first fraction. Since $y^4 \div y^4 = 1$, the expression simplifies to $\frac{1}{-2x^2} \cdot -xy$
2
Step 2: Multiply the two terms together. Multiply the numerators: $1 \cdot (-xy) = -xy$; multiply the denominators: $-2x^2 \cdot 1 = -2x^2$. The expression becomes $\frac{-xy}{-2x^2}$
3
Step 3: Simplify the resulting fraction. First, cancel out the negative signs (a negative divided by a negative is positive), getting $\frac{xy}{2x^2}$. Then cancel the common factor $x$ in the numerator and denominator: $x \div x = 1$, and $x^2 \div x = x$. This leaves the simplified expression $\frac{1}{2x}$
Knowledge Points Involved
1
Simplifying Rational Expressions
This involves canceling common factors in the numerator and denominator of a fraction. Common factors can be numerical coefficients or variable terms with the same base, using the rule $\frac{a^m}{a^n}=a^{m-n}$ for non-zero $a$. It is used to reduce rational expressions to their simplest form.
2
Multiplication of Monomials
When multiplying monomials, multiply the numerical coefficients together and multiply like variable bases by adding their exponents, following the rule $a^m \cdot a^n = a^{m+n}$. This applies to both positive and negative coefficients, where the product of two negative terms is positive.
3
Rules of Exponents for Division
For non-zero bases $a$, $\frac{a^m}{a^n}=a^{m-n}$. When $m=n$, this simplifies to $a^0=1$, which is used to cancel out identical variable terms in the numerator and denominator of a rational expression.
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