Problem 17 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( x^8 y^{10} \).

Step 1: Expand the Power

Apply the power of a power rule \( (a^m)^n = a^{m imes n} \) to the expression \( (x^2 y^3)^3 \). This gives \( x^{2 imes 3} imes y^{3 imes 3} = x^6 y^9 \).

Step 2: Rewrite the Expression

Replace the expanded form in the expression \( (x^2 y^3)^3 imes (x^2 y) \) with its simplified form from Step 1: \( x^6 y^9 imes (x^2 y) \).

Step 3: Multiply Like Bases

Combine the terms by adding the exponents of like bases. For \( x \), use the exponents \( 6 \) and \( 2 \) to get \( x^{6+2} = x^8 \).For \( y \), use the exponents \( 9 \) and \( 1 \) to get \( y^{9+1} = y^{10} \).

Step 4: Write the Final Simplified Expression

The simplified expression, combining all terms, is \( x^8 y^{10} \).

Key Concepts

Power of a Power Rule

In algebra, the power of a power rule is a handy tool for simplifying expressions with exponents. It states that \[ (a^m)^n = a^{m \times n} \]. This means when you have an expression where a power is raised to another power, you can multiply the exponents.

In our exercise, we start with \((x^2 y^3)^3\). Both the variables \(x\) and \(y\) have exponents that are raised to another power.

• For \(x^2\), the exponent \(2\) is raised to the \(3\)rd power, so we calculate \(x^{2 \times 3} = x^6\).
• For \(y^3\), the exponent \(3\) is also raised to the \(3\)rd power, resulting in \(y^{3 \times 3} = y^9\).

This approach helps simplify complex exponential expressions into more manageable forms, making them easier to work with in further calculations.

Simplifying Expressions

Simplifying algebraic expressions involves combining like terms and creating a more compact form of the expression.

After applying the power of a power rule, the expression \((x^2 y^3)^3 \times (x^2 y)\) becomes \(x^6 y^9 \times (x^2 y)\).

Simplification means reducing it to its simplest form:

• Identify like terms, which in this case are the terms involving the same base variables \(x\) and \(y\).
• Combine them by adding their exponents.

In our example, \(x^6\) and \(x^2\) can be combined, as can \(y^9\) and \(y\). This results in a simpler expression \(x^8 y^{10}\). Simplified expressions are often easier to work with and can make solving equations more straightforward.

Exponents Manipulation

Manipulating exponents is crucial for dealing with polynomial expressions and making calculations more straightforward.
Exponents tell you how many times a number, or base, is multiplied by itself. When expressions with exponents are multiplied together, you typically add the exponents for bases that are the same, following the rule:\[ a^m \times a^n = a^{m+n} \].
In our exercise:

• For the base \(x\), the exponents 6 and 2 (from \(x^6\) and \(x^2\)) are added, resulting in \(x^{6+2} = x^8\).
• For the base \(y\), we add 9 and 1 (from \(y^9\) and \(y\)), leading to \(y^{9+1} = y^{10}\).

By understanding this rule, you can tackle various algebraic problems more effectively. It’s all about knowing when to add, subtract, or multiply exponents based on the operations performed on the algebraic terms.