Problem 30 - Beginning and Intermediate Algebra

Question

Simplify. $$(2 x)^{5}(3 x)^{2}$$

Step-by-Step Solution

Answer

The short answer based on the step-by-step solution is: \( 288x^7 \).

Step 1: Expand the expression using exponent rules

First, let's use the exponent rules to rewrite the expression. By the rules of exponents, we know that \((ab)^n = a^nb^n\), so we can apply this to our expression: \((2x)^5(3x)^2 = (2^5x^5)(3^2x^2)\)

Step 2: Simplify the constants

Now, we will simplify the constants raised to their exponents: \(2^5 = 32\) and \(3^2 = 9\) So we have: \( (32x^5) (9x^2) \)

Step 3: Combine the terms

To finish, we'll multiply the constants and add the exponents of the x terms: \(32 \cdot 9 = 288\) The sum of the exponents is: \(5 + 2 = 7\) Thus, the simplified expression is: \( 288x^7 \)

Key Concepts

Simplification

Simplification is a crucial step when dealing with expressions, especially in algebra. It's all about making an expression easier to understand or use. Here,

• we break down complex expressions into simpler forms, following specific rules and operations.

By applying exponent rules, we can rewrite or break expressions into basic, comprehensible parts. For example, in our problem:

• We first expanded \[(2x)^5(3x)^2\] into \[(2^5x^5)(3^2x^2)\].

We simplified numerical bases

• \[2^5 = 32\] and \[3^2 = 9\].

Finally, by multiplying the numerical coefficients 32 and 9, and adding the exponents of like terms, simplification makes the original problem more straightforward, achieving the desired result of \[288x^7\].

Multiplying Exponents

Multiplying exponents can initially seem complex, but it's quite manageable once we apply the right rules. When multiplying similar bases, you simply add the exponents. For example, the general rule is:

• \[a^m imes a^n = a^{m+n}\].

In our case of \[(32x^5)(9x^2)\],

• we keep the base 'x' and add the exponents:
• \[x^5 imes x^2 = x^{5+2} = x^7\].

Using this rule, we efficiently combine powers of the same base, leading to a single base with a sum of the original exponents. This simplifies the computation and makes problems less complicated.

Exponentiation

Exponentiation is the process of raising a number to a power. It's a shortcut to representing repeated multiplication of the same base, noted as:

• \[b^n = b \times b \times ... \times b\], for n times.

Let's review the process using our problem. Consider \[(2x)^5\].

• This represents: \[2^5 \times x^5\].

Exponentiation applies separately to each part within the parentheses.

• This also holds for \[(3x)^2\], equating to: \[3^2 \times x^2\].

Understanding each component individually allows for stepwise simplification, eventually combining them to derive the expression \[288x^7\].

Algebra Basics

Algebra is the foundation of higher mathematics, using symbols to represent numbers and operations. It allows us to generalize mathematical ideas and solve complex problems. Expressions like \[(2x)^5(3x)^2\] are quintessential examples of algebra in action.

• Here, operations such as multiplication, exponentiation, and simplification highlight algebra's core principles.

One critical skill in algebra is understanding how to manipulate equations and expressions using:

• Proper order of operations (PEMDAS/BODMAS).
• Rules for exponents and coefficients.

Understanding these tools and rules transforms daunting tasks into manageable operations, like simplifying \[288x^7\].Arming yourself with these algebra basics makes mathematical expressions less intimidating and more engaging.