Problem 49 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(-3 a b^{3}\right)^{4}$$

Step-by-Step Solution

Answer

The solution is \(81a^4b^{12}\).

Step 1: Understand the Problem

We need to raise the given monomial \( (-3ab^3) \) to the fourth power.

Step 2: Apply the Exponent Rule

Recall that \( (xy)^n = x^n y^n \). We apply this rule to \((-3ab^3)^4\), separating the components: \((-3)^4\), \(a^4\), and \((b^3)^4\).

Step 3: Calculate \((-3)^4\)

Calculate \((-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 81\).

Step 4: Calculate \(a^4\)

Simply note \(a^4\) remains \(a^4\), since \(a\) is already to the power of one.

Step 5: Calculate \((b^3)^4\)

Apply the power of a power rule \((x^m)^n = x^{mn}\). So, \((b^3)^4 = b^{3 \times 4} = b^{12}\).

Step 6: Combine the Results

Combine the results from earlier steps: \(81a^4b^{12}\).

Key Concepts

Exponent Rules

Exponent rules are the fundamental guidelines for simplifying expressions involving powers. Exponents allow us to express repeated multiplication in a compact form.

When we talk about exponent rules, we essentially refer to a set of valuable properties and equations, such as:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers:\( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power:\( (a^m)^n = a^{mn} \)
• Power of a Product:\( (ab)^n = a^n b^n \)

These are just a few of the rules that help simplify expressions when you work with powers. By applying these rules, complicated expressions become far more manageable.

Monomials

Monomials are the simplest forms of polynomial expressions. A monomial consists of just one term, which can include:

• a constant
• a variable
• or a product of constants and variables

For example, the expression \(-3ab^3\) from our problem is a monomial, comprised of the constant \(-3\), variable \(a\), and expression \(b^3\). These components are all multiplied together to form a single coherent unit.

Monomials can be raised to powers, multiplied, and divided according to exponent rules, making them versatile and fundamental in algebra.

Power of a Product Rule

The Power of a Product Rule is a key exponent rule that simplifies expressions where a product is raised to a power. This rule can be stated as:\[(ab)^n = a^n \cdot b^n\]When you have a product within the parentheses and the entire expression is raised to a power, you can distribute the power to each factor within the product separately.

In our example, we have \((-3ab^3)^4\).According to the Power of a Product Rule, we can split it as follows:

• Raise \(-3\) to the fourth power
• Raise \(a\) to the fourth power
• And raise \((b^3)\) to the fourth power, which then requires the Power of a Power Rule for simplification

This approach systematically breaks down complex expressions to more digestible calculations.

Power of a Power Rule

The Power of a Power Rule helps when you have an exponent raised to another exponent. This important rule is expressed as:\[(x^m)^n = x^{m \cdot n}\]This means if a power is raised to another power, you multiply the exponents.

In our example, consider the term \((b^3)^4\).By using the Power of a Power Rule, you determine:

• Multiply the inner exponent (3) by the outer exponent (4)
• This equates to \(b^{3\times4}\)
• Which simplifies to \(b^{12}\)

This rule helps to consolidate complex power structures into simpler, workable terms, making your algebraic manipulations smoother and easier to handle.