Problem 40 - Algebra 2

Question

Simplify each expression. \(\frac{\left(-5 m n^{2}\right)^{3}}{5 m^{2} n^{4}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-25mn^2\).

Step 1: Apply Power to the Numerator

First, apply the exponent of 3 to each term inside the parentheses in the numerator: \[(-5mn^2)^3 = (-5)^3 \times (m)^3 \times (n^2)^3 = -125m^3n^6\]

Step 2: Write Out the Expression

Rewrite the expression with the simplified numerator from Step 1:\[\frac{-125m^3n^6}{5m^2n^4}\]

Step 3: Simplify the Coefficients

Divide the numerical coefficient in the numerator by the one in the denominator:\[\frac{-125}{5} = -25\]So the expression becomes:\[-25\frac{m^3n^6}{m^2n^4}\]

Step 4: Simplify the Variables

Apply the quotient rule of exponents to the variables:- For \(m\), subtract the exponent in the denominator from the exponent in the numerator: \[m^{3-2} = m^1 = m\]- For \(n\), subtract the exponent in the denominator from the exponent in the numerator:\[n^{6-4} = n^2\]Thus, the expression is simplified to:\[-25mn^2\]

Key Concepts

Exponents

An exponent indicates how many times a number, known as the base, is multiplied by itself. In the expression \((-5mn^2)^3\), the base \((-5mn^2)\) is being raised to the power of 3. This means you multiply \((-5mn^2)\) by itself three times.

• The exponent 3 transforms each factor inside the parentheses.
• Apply the exponent to each component: \((-5)^3, m^3, \) and \((n^2)^3\).

Breaking this down:

• \((-5)^3 = -125\)
• \(m^3\)
• For \(n^2\), the power is applied, resulting in \(n^6\).

Once combined, these give \(-125m^3n^6\). Using exponents makes calculations quicker and expressions easier to manage.

Simplifying Expressions

Simplifying an expression often involves combining like terms and reducing complexity, ultimately presenting the expression in the most efficient way. For example, after calculating the numerator in our given exercise, the expression became \(\frac{-125m^3n^6}{5m^2n^4}\).To simplify:

• Look for common factors in the coefficients, (here: -125 and 5).
• Utilize the laws of exponents to tackle variables.

The simplified coefficient emerged from dividing -125 by 5, which is -25. This simplification step makes further reduction work focused on the variables left in the expression. By simplifying, we achieve a more comprehensible and concise form of the original expression.

Quotient Rule

The quotient rule of exponents assists in simplifying expressions that involve division of similar bases. The rule states: when dividing two powers with the same base, you subtract the exponents. Mathematically, this is expressed as \[(a^m)/(a^n) = a^{m-n}.\]Applying this rule here:

• For \(m^3/m^2\), subtract the exponent in the denominator from that in the numerator \(m^{3-2} = m^1 = m\).
• Similarly, for \(n^6/n^4\), subtract the exponents to get \(n^{6-4} = n^2\).

The quotient rule drastically simplifies the expression, removing unnecessary components while preserving mathematical accuracy. Its utility in handling expressions with high exponents can significantly reduce calculation time.

Numerators and Denominators

Numerators and denominators are fundamental terms in fraction expressions.

• Numerator: the top part of a fraction, which in this exercise started as \(-125m^3n^6\).
• Denominator: the bottom part of a fraction, initially \(5m^2n^4\).

When simplifying, we deal with each component separately and together's effect, especially focusing on the division resulting from simplification. By dividing both components and applying rules such as the quotient rule, the expression transforms.

• Separate numerical simplification worked for coefficients \(-125/5\).
• Variable simplification used the same exponent rule resulting in \(-25mn^2\).

This understanding of numerators and denominators as distinct yet related components of a fraction is essential in algebraic operations.