Problem 11 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(x^{4} \cdot x^{2} y^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^6 y^3\).

Step 1: Identify Like Terms

The expression is given as \(x^4 \cdot x^2 y^3\). Notice that \(x^4\) and \(x^2\) are like terms because they have the same base \(x\). The term \(y^3\) will remain as it is since there are no other \(y\) terms to combine with.

Step 2: Apply the Product of Powers Rule

Use the product of powers rule, which states that when multiplying like bases, you add their exponents: \(x^a \cdot x^b = x^{a + b}\). Thus, apply this to \(x^4 \cdot x^2\), giving \(x^{4+2} = x^6\).

Step 3: Write the Simplified Expression

Now that we have combined the like terms, the simplified expression becomes \(x^6 y^3\).

Key Concepts

Product of Powers Rule

The **Product of Powers Rule** is a fundamental principle in algebra that helps you simplify expressions involving exponents. When you encounter terms with the same base being multiplied, such as \(x^4 \cdot x^2\), you can use this rule to simplify the expression more easily. The rule is straightforward: for two exponents with the same base, you simply add the exponents together. So, for our example of \(x^4 \cdot x^2\), you add the exponents 4 and 2 together. This gives you a new term: \(x^{4+2} = x^6\). The base \(x\) remains the same, while the exponents are summed up.

This rule emphasizes efficiency in dealing with powers and ensures that expressions are presented in a simplified manner, helping you solve equations and compare expressions quickly.

Like Terms

**Like terms** are terms within an algebraic expression that have the same variable raised to the same power, though they might have different coefficients. Recognizing like terms is crucial because they can be combined, making expressions simpler. For example, in the expression \(x^4 \cdot x^2 y^3\), the like terms are \(x^4\) and \(x^2\), because both terms contain the variable \(x\).

When you spot like terms, you can use rules like the Product of Powers Rule to combine them efficiently. This helps in reducing complex expressions to their simplest form. Remember, terms with different bases or differing exponents on the same base are not considered like terms, and cannot be combined in this manner. Practicing the identification of like terms makes algebraic manipulations much more intuitive.

Simplifying Expressions

**Simplifying Expressions** is the process of making an algebraic expression as straightforward as possible. It involves combining like terms and reducing the complexity of the expression through various algebraic rules. In our example of \(x^4 \cdot x^2 y^3\), the simplification process is as follows:

• First, identify and combine any like terms, such as \(x^4\) and \(x^2\), using relevant rules like the Product of Powers Rule.

After applying this rule, we simplified \(x^4 \cdot x^2\) to \(x^6\).

• Once like terms are combined or eliminated, write the remaining expression in its simplest form.

In our case, since no other terms with the base \(y\) exist, \(y^3\) remains unchanged.

The final, simplified expression is \(x^6 y^3\). By condensing the expression to its simplest form, we make it easier to work with for further mathematical operations or comparisons. Simplifying expressions in algebra is like tidying up a cluttered room, making it easier to find and use what you need.