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

The question asks you to perform algebraic simplification on the given expression (-15m^5n^-7)/(3m^-2n^-3). Simplification in this context typically involves combining like terms, using the laws of exponents to simplify powers, and reducing fractions to their simplest form. It requires the you to carry out the necessary steps to simplify the rational expression by handling both the numerical and the variable parts, with attention paid to the negative exponents which indicate reciprocal powers.

$\frac{- 15 m^{5} n^{- 7}}{3 m^{- 2} n^{- 3}}$

Answer

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Solution:

Step 1:

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

Step 2:

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

Step 2.1:

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

Step 2.2:

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

Step 2.3:

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

Step 3:

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

Step 4:

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

Step 4.1:

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

Step 4.2:

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

Step 4.3:

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

Step 5:

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

Step 5.1:

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

Step 5.2:

Cancel out the common factors.

Step 5.2.1:

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

Step 5.2.2:

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

Step 5.2.3:

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

Step 6:

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

Knowledge Notes:

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.
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.
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 $c \neq 0$ .
Simplifying Expressions: When simplifying algebraic expressions, it is important to combine like terms and apply the appropriate algebraic rules to reach the simplest form.
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.