Simplify (-15m^5n^-7)/(3m^-2n^-3)

Solution:

Step 1:

Apply the negative exponent rule $b^{-n} = \frac{1}{b^n}$ to reposition $n^{-7}$ to the denominator. Thus, we get $\frac{-15m^5}{3m^{-2}n^{-3}n^7}$.

Step 2:

Combine $n^{-3}$ and $n^7$ by summing their exponents.

Step 2.1:

Reposition $n^7$ to combine with $n^{-3}$: $\frac{-15m^5}{3m^{-2}(n^7n^{-3})}$.

Step 2.2:

Utilize the exponent rule $a^ma^n = a^{m+n}$ to merge the exponents: $\frac{-15m^5}{3m^{-2}n^{7-3}}$.

Step 2.3:

Calculate $7 - 3$ to simplify the exponent of $n$: $\frac{-15m^5}{3m^{-2}n^4}$.

Step 3:

Reposition $m^{-2}$ to the numerator using the rule $\frac{1}{b^{-n}} = b^n$: $\frac{-15m^5m^2}{3n^4}$.

Step 4:

Multiply $m^5$ by $m^2$ by adding their exponents.

Step 4.1:

Reposition $m^2$ to combine with $m^5$: $\frac{-15(m^2m^5)}{3n^4}$.

Step 4.2:

Apply the exponent rule $a^ma^n = a^{m+n}$ to merge the exponents: $\frac{-15m^{2+5}}{3n^4}$.

Step 4.3:

Calculate $2 + 5$ to simplify the exponent of $m$: $\frac{-15m^7}{3n^4}$.

Step 5:

Eliminate the common factor between $-15$ and $3$.

Step 5.1:

Extract the factor of $3$ from $-15m^7$: $\frac{3(-5m^7)}{3n^4}$.

Step 5.2:

Cancel out the common factors.

Step 5.2.1:

Factor out $3$ from $3n^4$: $\frac{3(-5m^7)}{3(n^4)}$.

Step 5.2.2:

Cancel the common factor of $3$: $\frac{\cancel{3}(-5m^7)}{\cancel{3}n^4}$.

Step 5.2.3:

Rewrite the simplified expression: $\frac{-5m^7}{n^4}$.

Step 6:

Place the negative sign in front of the fraction: $-\frac{5m^7}{n^4}$.

Knowledge Notes:

1. Negative Exponent Rule: For any nonzero number $b$ and any integer $n$, $b^{-n} = \frac{1}{b^n}$. This rule allows us to transform negative exponents into positive ones by moving the base to the opposite side of the fraction.

2. Power Rule for Exponents: When multiplying two powers with the same base, you can add the exponents: $a^m \cdot a^n = a^{m+n}$. This is used to simplify expressions with the same base.

3. Simplifying Fractions: When simplifying fractions, any common factors in the numerator and denominator can be canceled out. This is based on the property that $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$ when $c \neq 0$.

4. Simplifying Expressions: When simplifying algebraic expressions, it is important to combine like terms and apply the appropriate algebraic rules to reach the simplest form.

5. Handling Negative Signs: Negative signs can be moved in front of a fraction or to the numerator, depending on the context and the need for simplification. The overall value of the expression remains unchanged.