Problem 27 - Elementary Algebra

Question

\(\left(a^{3} b\right)^{2}\)

Step-by-Step Solution

Answer

\(a^6 b^2\)

Step 1: Understand the given expression

The expression given is \(\big( a^{3} b \big)^{2}\). This needs to be simplified.

Step 2: Apply the power rule

Use the power rule which states that \( (x^m)^n = x^{(m \times n)} \). Apply this to both the bases inside the parentheses.

Step 3: Distribute the exponent

Distribute the exponent outside the parentheses to both the terms inside. So, \((a^3 b)^2 = (a^3)^2 \times (b)^2\).

Step 4: Simplify the terms

Now, simplify each term individually. \((a^3)^2 = a^{3 \times 2} = a^6\) and \(b^2 = b^2\).

Step 5: Combine the simplified terms

Combine the simplified terms to get \(a^6 \times b^2\). Therefore, the simplified form of the given expression is \(a^6 b^2\).

Key Concepts

Power Rule

When we talk about the power rule in exponentiation, it's all about simplifying expressions involving exponents. The power rule states that when raising a power to another power, you multiply the exponents. This is represented mathematically as \((x^m)^n = x^{(m \times n)}\).

For example, given the expression \((a^3 b)^2\), you can use the power rule to simplify it. Start by applying the rule separately to each term within the parentheses.

This process transforms \((a^3 b)^2\) into \((a^3)^2 \times (b)^2\). Notice that each exponent inside the parentheses is being raised to the power outside.

Exponent Distribution

Exponent distribution is a fundamental concept where the exponent outside the parenthesis is distributed to every term inside. Specifically, in the expression \((a^3 b)^2\), the exponent 2 is given to both \(a^3\) and \(b\).

Using exponent distribution, the expression \((a^3 b)^2\) becomes \((a^3)^2 \times (b)^2\). This means you handle each term individually before putting them back together.

Remember:

• Distribute the exponent to terms multiplied inside the parenthesis.
• Break down the expression step-by-step to avoid confusion.

Once distribution is done, the next step is simplifying each part.

Simplifying Exponents

Simplifying exponents involves combining and reducing them to their simplest form. After distributing the exponent in \((a^3 b)^2\), you get \((a^3)^2 \times (b)^2\). Let's simplify each part step by step:

First, take \((a^3)^2\):

• Use the power rule to multiply the exponents: \(a^{3 \times 2} = a^6\).

Next, simplify \((b)^2\):

• This simply becomes \b^2\ because it was already raised to the power 2.

Now, combine both simplified terms:

• The final answer becomes \a^6 \times b^2\, or simply \a^6 b^2\.

Understanding these steps helps in making exponentiation problems much easier!