Problem 37 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ 4 x \cdot\left(x \cdot x^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(4x^{9}\).

Step 1: Analyze the expression

The given expression \(4x \cdot (x \cdot x^{3})^{2} \) has an inner expression \(x \cdot x^{3}\) raised to the power of 2. We also have a 4x term that is being multiplied to this whole expression.

Step 2: Simplify the inner expression

Make use of the multiplication rule when the bases are the same. Here, the base, x, is the same. Applying the rule \(x \cdot x^{3} = x^{1+3} = x^{4}\). So the expression now simplifies to: \(4x \cdot (x^{4})^{2}\).

Step 3: Apply power of a power rule and simplify further

One can notice that there is an opportunity to apply the power of a power rule on \(x^{4})^{2}\). According to this rule, when an exponent is raised to another exponent, you multiply the exponents. Applying this rule \(x^{4})^{2} = x^{4*2} = x^{8}\). So, the expression now simplifies to: \(4x \cdot x^{8}\).

Step 4: Apply multiplication rule with same bases once more

Finally, apply the multiplication rule for the same base once more on \(4x \cdot x^{8} = 4x^{1+8} = 4x^{9}\). Considering that the 4 is a constant, we must leave it as it is.

Key Concepts

Powers

Understanding powers are crucial when dealing with algebraic expressions. A power indicates that a number, known as the base, is multiplied by itself a certain number of times. For instance, in the expression \(x^3\), \(x\) is the base, and 3 is the exponent. The expression means "\(x\) multiplied by itself 3 times".

• The result of multiplying the base by itself is called a power.
• Powers are often used to simplify large multiplication problems.

Recognizing and working with powers is fundamental to simplifying expressions involving exponents.

Multiplication of Exponents

Multiplication of exponents occurs when you multiply terms that have similar bases. If the bases match, you can simply add the exponents. This is known as the multiplication rule of exponents, expressed as \(x^a \cdot x^b = x^{a+b}\).Here's how you can apply this:

• Identify terms with the same base.
• Add their exponents together.

For example, in the expression \(x \cdot x^3\), you add the exponents 1 and 3 to get \(x^4\). It's a helpful rule to swiftly simplify multiplication of similar bases.

Simplifying Expressions

Simplifying expressions is a common practice in algebra, aimed at reducing expressions to their simplest form. This often involves combining like terms and applying rules of exponents effectively.

• Identify any expressions or terms that can be combined.
• Use exponent rules such as multiplication of exponents or power of a power to simplify.

For example, in the expression \((x^4)^2\), you apply the power of a power rule which states you multiply the exponents, resulting in \(x^8\). The overall goal of simplifying is to make the expression easier to interpret and solve.