Problem 43 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(a b^{2} c^{3}\right)^{6}$$

Step-by-Step Solution

Answer

The expressed monomial is \(a^6 b^{12} c^{18}\).

Step 1: Apply the Power to Each Part of the Monomial

The given monomial is \( (a b^2 c^3)^6 \). According to the law of exponents, when you raise a power to another power, you multiply the exponents. Therefore, distribute the exponent 6 to each part of the monomial separately: \[(a^1)^6 , (b^2)^6, (c^3)^6\].

Step 2: Multiply Exponents

For each of the factors, multiply the exponents: \(a^1\) becomes \(a^{1 imes 6} = a^6\), \(b^2\) becomes \(b^{2 imes 6} = b^{12}\), \(c^3\) becomes \(c^{3 imes 6} = c^{18}\).

Step 3: Write the Final Expression

Combine the results from Step 2 to write the fully expanded form of the monomial.Thus, the expression \((a b^2 c^3)^6\) becomes: \[a^6 b^{12} c^{18} \].

Key Concepts

Monomial

A monomial is a mathematical expression composed of a single term. It consists of coefficients and variables raised to non-negative integer powers. For instance, in the expression \( a b^2 c^3 \), each variable, \(a\), \(b\) and \(c\), is part of the monomial and each has its own exponent. An important aspect of monomials is that they don't have any addition or subtraction operators splitting the term. This means no separate terms are cashing in on the sum or difference here—just a solitary term doing its own thing!

• The coefficient in a monomial is the number in front of the variables, which in this case implies \( 1 \) as there is no visible number.
• Exponents are crucial as they show how many times to multiply the variable by itself.

Understanding monomials is key in algebra as they are the building blocks for more complex expressions, and you'll often find yourself raising them to powers, as in the given exercise.

Power Rule

The power rule is a fundamental law of exponents that states when you raise a power to another power, you multiply the exponents. This rule simplifies the process of working with powers and is essential for manipulating expressions involving exponents.

Using the Power Rule

When applying the power rule to a monomial like \((a b^2 c^3)^6\), you distribute the outer exponent to each part inside the bracket. This means:

• For \(a^1\), apply the power: \((a^1)^6 = a^{1 \times 6} = a^6\).
• For \(b^2\), apply the power: \((b^2)^6 = b^{2 \times 6} = b^{12}\).
• For \(c^3\), apply the power: \((c^3)^6 = c^{3 \times 6} = c^{18}\).

This results in the expression being transformed to \(a^6 b^{12} c^{18}\). By using the power rule, calculations become more streamlined, leading to more complex manipulations with ease.

Law of Exponents

The law of exponents encompasses various rules that govern operations involving exponents. These rules help simplify expressions and solve exponential equations efficiently. In the exercise, we mainly focus on two key laws:

Product of Powers Rule

- This states that to multiply two powers with the same base, you add the exponents: \(x^m \times x^n = x^{m+n}\).

Power of a Power Rule

- As used in the exercise, this specifies that when raising a power to a new power, you multiply the exponents: \((x^m)^n = x^{m \times n}\).These laws allow you to break down complicated exponent expressions into manageable pieces. For example, raising the entire monomial \((a b^2 c^3)^6\) involves applying these laws efficiently to obtain the expanded form \(a^6 b^{12} c^{18}\). This expanded expression is easy to work with and crucial in solving more complex algebraic problems.