Problem 45 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\((2a^2b^3)^6 = 64a^{12}b^{18}\).

Step 1: Understanding the Problem

We need to raise the monomial \(2a^2b^3\) to the power of 6. This involves applying the power to each factor in the monomial.

Step 2: Applying the Power of a Product Rule

The power of a product can be distributed to each factor in the product. This means \((xy)^n = x^n y^n\). So, \((2a^2b^3)^6 = 2^6 (a^2)^6 (b^3)^6\).

Step 3: Simplifying Each Factor

Now, simplify each factor using the rules of exponents: - \(2^6 = 64\)- \((a^2)^6 = a^{2 \times 6} = a^{12}\)- \((b^3)^6 = b^{3 \times 6} = b^{18}\)

Step 4: Writing the Final Expression

Combine the simplified factors to write the final expression: \((2a^2b^3)^6 = 64a^{12}b^{18}\).

Key Concepts

Monomial

A monomial is a mathematical expression consisting of a single term which can be a constant, a variable, or the product of constants and variables. In our exercise, the monomial is \(2a^2b^3\). Here, we have:

• Coefficient: The constant number in front of the variables, which is 2 in this case.
• Variables: These are "\(a\)" and "\(b\)" that are raised to powers.
• Exponents: The powers to which the variables are raised, "\(a^2\)" means "a squared," and "b^3" means "b cubed."

Understanding monomials is vital because they are the building blocks of more complex polynomials. It is essential to master how to manipulate these expressions under various mathematical operations like multiplication, division, and exponentiation.

Power of a Product

The 'power of a product' is a crucial concept particularly useful when raising monomials that consist of multiple factors to a power. In our example, we need to compute \((2a^2b^3)^6\). The power of a product states that when a product is raised to an exponent, each factor in the product is raised to that exponent. This can be easily expressed as:\[ (xy)^n = x^n y^n \]For our exercise, this rule transforms our expression into \(2^6 (a^2)^6 (b^3)^6\). It allows us to distribute the exponent over each factor separately, making it much easier to handle complex expressions. Remember, the power applies not just to variables but to any product of numbers and/or variables.

Exponent Rules

Exponent rules are fundamental to solving problems involving powers. They provide a set of guidelines to simplify expressions efficiently. Let's apply these rules to our exercise step-by-step:

• Power of a Power Rule: This rule states that when raising a power to another power, you multiply the exponents. It is represented by \((x^m)^n = x^{m \,* n}\). For example, \((a^2)^6 = a^{12}\) and \((b^3)^6 = b^{18}\).
• Power of a Product Rule: We already applied this in the previous section, allowing us to handle expressions like \((2a^2b^3)^6\) by raising each component to the 6th power.

By applying these rules, we simplify \(2^6 = 64\), \(a^{12}\), and \(b^{18}\), leading us to the final expression of \(64a^{12}b^{18}\). These rules make it efficient to handle even more complex expressions systematically.