Problem 61 - Elementary Algebra

Question

\((6 x)^{2}(2 x)^{3}\)

Step-by-Step Solution

Answer

288x^{5}

Step 1: Identify the terms to be combined

The expression \((6x)^{2}(2x)^{3}\) contains two separate terms, \(6x\) and \(2x\), each raised to their own powers.

Step 2: Apply the exponent rules to each term

Raise each term to its respective power:The first term: \((6x)^{2} = 6^{2} \times x^{2} = 36x^{2}\)The second term: \((2x)^{3} = 2^{3} \times x^{3} = 8x^{3}\)

Step 3: Multiply the simplified terms

Multiply the results of the two simplified terms together: \((36x^{2})(8x^{3})\).

Step 4: Combine the constants and variables

Multiply the constants and the variables separately:Constants: \(36 \times 8 = 288\)Variables: \(x^{2} \times x^{3} = x^{(2+3)} = x^{5} \)Thus, the combined result is \(288x^{5}\).

Key Concepts

Exponent Rules

Understanding exponent rules is key to solving expressions like \( (6x)^{2}(2x)^{3} \). Exponents indicate how many times a number, known as the base, is multiplied by itself. Here are the basic rules you'll need:

• Product of Powers Rule: When you multiply two powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Power of a Product Rule: Distribute the exponent to each factor in the product: \((ab)^m = a^m \times b^m\).

In the provided exercise, we use these rules to simplify each term before combining them.

Algebraic Expressions

An algebraic expression is a combination of variables, constants, and arithmetic operations. In the expression \( (6x)^{2}(2x)^{3} \), it consists of:

• Constants: These are numbers without any variables (6 and 2).
• Variables: These are symbols that represent unknown values (x in this case).
• Exponents: Numbers that indicate how many times the variable is to be multiplied by itself (2 and 3).

To simplify algebraic expressions with exponents, apply exponent rules to each term. For example, \(6^2 \times x^2 \) becomes \(36x^2 \), which involves both constants and variables raised to their exponents.

Combining Like Terms

Combining like terms is an essential part of simplifying algebraic expressions. 'Like terms' are terms that contain the same variables raised to the same power. Here's how to combine them:

• Identify Like Terms: In \( (36x^2)(8x^3) \), 36 and 8 are constants, and \(x^2 \) and \( x^3 \) are like terms because they have the same base.
• Multiply Constants Separately: Calculate \36 \times 8 = 288 \.
• Add Exponents for Variables: Since the base (x) is the same, add the exponents: \ x^{2+3} = x^5 \.
• Combine Results: The final simplified expression is \288x^5 \.

By identifying and combining like terms correctly, you can simplify complex expressions into a more manageable form.