Simplify (-5y^10)/(30y^5)

The problem is asking to simplify a fraction that involves both numerical and variable terms. Specifically, the fraction (-5y^10)/(30y^5) includes variables raised to powers (y^10 and y^5) and also contains coefficients (-5 and 30). The process of simplifying would involve reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common factor, and applying the laws of exponents for the variables to simplify y^10/y^5.

$\frac{- 5 y^{10}}{30 y^{5}}$

Answer

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Solution:

Step 1: Simplify the numerical coefficients.

Identify and divide out the common numerical factor from both the numerator and the denominator.

Step 1.1: Extract the factor of 5 from the numerator.

Rewrite the numerator by factoring out 5: $\frac{- 5 y^{10}}{30 y^{5}} = \frac{5 (- y^{10})}{30 y^{5}}$ .

Step 1.2: Simplify the common numerical factors.

Factor out 5 from the denominator.

Step 1.2.1: Factor out 5 from the denominator.

Express the denominator as a product of 5 and another term: $\frac{5 (- y^{10})}{5 (6 y^{5})}$ .

Step 1.2.2: Cancel out the common numerical factor.

Remove the common factor of 5 from both the numerator and the denominator: $\frac{5 (- y^{10})}{5 (6 y^{5})}$ .

Step 1.2.3: Rewrite the simplified expression.

After canceling out the common factor, the expression becomes: $\frac{- y^{10}}{6 y^{5}}$ .

Step 2: Simplify the variable factors.

Identify and divide out the common variable factor from both the numerator and the denominator.

Step 2.1: Extract the factor of $y^{5}$ from the numerator.

Rewrite the numerator by factoring out $y^{5}$ : $\frac{- y^{10}}{6 y^{5}} = \frac{y^{5} (- y^{5})}{6 y^{5}}$ .

Step 2.2: Simplify the common variable factors.

Factor out $y^{5}$ from the denominator.

Step 2.2.1: Factor out $y^{5}$ from the denominator.

Express the denominator as a product of $y^{5}$ and another term: $\frac{y^{5} (- y^{5})}{y^{5} \cdot 6}$ .

Step 2.2.2: Cancel out the common variable factor.

Remove the common factor of $y^{5}$ from both the numerator and the denominator: $\frac{y^{5} (- y^{5})}{y^{5} \cdot 6}$ .

Step 2.2.3: Rewrite the simplified expression.

After canceling out the common factor, the expression becomes: $\frac{- y^{5}}{6}$ .

Step 3: Adjust the negative sign.

Place the negative sign in front of the fraction to get the final result: $- \frac{y^{5}}{6}$ .

Knowledge Notes:

The problem involves simplifying a rational expression with both numerical and variable factors. Here are the relevant knowledge points:

Common Factor: When simplifying fractions, any common factors in the numerator and denominator can be canceled out. This includes both numerical and variable factors.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions. In this case, we factored out common numerical and variable factors from the numerator and denominator.
Exponent Rules: When dividing terms with the same base, subtract the exponents. For example, $y^{m} / y^{n} = y^{m - n}$ .
Negative Signs: A negative sign can be moved in front of a fraction without changing the value of the expression. For example, $- \frac{a}{b} = \frac{- a}{b} = \frac{a}{- b}$ .
Simplification: The goal of simplification is to rewrite an expression in its simplest form, which means reducing it to the fewest terms possible and removing any common factors.

In this problem, we used these principles to simplify the given expression by canceling out common numerical and variable factors and adjusting the negative sign to obtain the final simplified form.