Problem 40 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ 2 x^{3} \cdot(3 x)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(18x^5\)

Step 1: Observe the given expression

Look at the given expression \(2 x^{3} \cdot(3 x)^{2}\). Observe that it is a product of two terms: \(2 x^{3}\) and \((3 x)^{2}\).

Step 2: Apply the power of a power rule

The second term \((3 x)^{2}\) includes a power of a power. This can be simplified using the rule \((a^m)^n = a^{m*n}\). This gives \((3 x)^{2} = 3^2 \cdot x^{2} = 9x^{2}\)

Step 3: Combine the terms

Now, replace \((3 x)^{2}\) in the initial expression with the result obtained in Step 2. Therefore, the expression becomes \(2 x^{3} \cdot 9x^{2}\). Using the rule for multiplying powers with the same base, we add the exponents and obtain \(2 \cdot 9 x^{3+2} = 18x^5\)

Key Concepts

Power of a Power Rule

In mathematics, the "Power of a Power Rule" is an essential tool for simplifying expressions involving exponents. This rule allows us to easily simplify terms where an exponent is raised to another exponent. For example, if you have

• an expression in the form \((a^m)^n\)

any base raised to an exponent that is further raised to another exponent, is simplified by multiplying the exponents. This means \((a^m)^n = a^{m \cdot n}\).

By applying this rule, we can simplify complex equations. In the original exercise, we see it in action with

• \((3x)^2\)

which breaks down to

• \((3)^2 \times (x)^2 = 9x^2\).

Breaking down the exponents helps to simplify expressions and solve problems more efficiently.

Multiplying Powers with the Same Base

When multiplying powers that have the same base, a useful shortcut comes into play. Instead of solving each power separately, we can use the rule that

• \(a^m \cdot a^n = a^{m+n}\).

Essentially, the exponents are added while the base remains constant. This rule tremendously simplifies calculations and helps us deal with large expressions more easily.

In the shared problem, once we have simplified

• \((3x)^2 = 9x^2\),

we need to combine it with the first part of the original expression:

• \(2x^3\).

Realizing both terms have the base \(x\), we use the rule to

• add the exponents \(3+2\) to get \(x^5\),

simplifying our expression to

• \(18x^5\).

This method streamlines calculations and can be a huge time-saver when working with algebraic expressions.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involves reducing the expression to its simplest form. The goal is to express the simplified form as compactly as possible without changing its value. This often involves collecting like terms and applying rules of exponents.

In our example, the simplification process begins by applying the power of a power rule to separate and individually deal with

• powers in terms like \((3x)^2\),

then combining resultant expressions that have similar bases.

After simplifying to

• \(9x^2\),

this integrates seamlessly with

• \(2x^3\),

using rules for multiplying similar bases to reduce complexity to

• \(18x^5\).

Simplifying not only makes expressions more manageable but can also uncover relationships within the variables that might not be immediately apparent. Mastering simplification techniques helps greatly in tackling more advanced mathematical challenges.