Problem 90 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(-3 x y^{2} a^{3}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(-27x^3y^6a^9\).

Step 1: Understand the Expression

The expression to simplify is \((-3xy^2a^3)^3\). This is a power raised to another power, indicating that we'll need to apply the laws of exponents.

Step 2: Apply the Power of a Product Rule

The rule \((ab)^n = a^n b^n\) allows us to distribute the exponent 3 to each factor inside the parentheses: \[(-3)^3 (x)^3 (y^2)^3 (a^3)^3\].

Step 3: Simplify Each Term

Calculate each term individually:- \((-3)^3 = (-3) imes (-3) imes (-3) = -27\).- \(x^3\) remains the same as we simply apply the power to the variable \(x\).- \((y^2)^3 = y^{2 imes 3} = y^6\).- \((a^3)^3 = a^{3 imes 3} = a^9\).

Step 4: Combine Simplified Terms

Put all the simplified terms together to get the final simplified expression:\[-27x^3y^6a^9\].

Key Concepts

Power of a Product Rule

The power of a product rule is one of the key laws of exponents that helps in simplifying expressions which involve products raised to a power. This rule states that

• \((ab)^n = a^n \times b^n\)

where \(a\) and \(b\) are numbers or variables, and \(n\) is the exponent.
In our example,

• \((-3xy^2a^3)^3\)

each factor inside the parentheses is raised to the exponent of 3. This results in:

• \((-3)^3\),
• \(x^3\),
• \((y^2)^3\),
• \((a^3)^3\).

By applying the power of a product rule, we simplify complex expressions by separating each part and raising them individually to the specified power.
Once each component is independently evaluated, we can piece them back together for the final result.

Simplifying Algebraic Expressions

Simplifying an algebraic expression involves rewriting it in a more concise form while maintaining its original value. This often means reducing the number of terms and using the laws of exponents to decrease the complexity of computations.
In the expression

• \((-3xy^2a^3)^3\),

we follow the steps using our exponent rules to break it down:

• Calculate \((-3)^3 = -27\) by multiplying \(-3\) by itself three times.
• Simply apply the power to the variable to get \(x^3\).
• For \((y^2)^3\), apply the power of a power rule: multiply the exponents to get \(y^{6}\).
• Similarly, \((a^3)^3\) becomes \(a^9\) using the same rule.

The resulting simplified expression is:

• \(-27x^3y^6a^9\).

By breaking down each part and carefully managing the individual components, the entire expression becomes easier to understand and handle.

Laws of Exponents

The laws of exponents govern the operations of expressions that have exponents, ensuring accuracy and simplicity in mathematical computations. These rules include

• The power of a product rule – already discussed,
• The product of powers rule – where \(a^m \times a^n = a^{m+n}\).
• The power of a power rule – used when we have an exponent raised to another exponent: \((a^m)^n = a^{m\times n}\).
• The negative exponent rule – which states \(a^{-n} = \frac{1}{a^n}\).

For the expression

• \((-3xy^2a^3)^3\)

we primarily used the power of a power and power of a product rules.

• Each part is broken down to its base and each base is raised to its respective power.
• By multiplying the exponents in \((y^2)^3\) and \((a^3)^3\),
• you see how the power of a power rule unfolds to simplify the expression to become \(y^6\) and \(a^9\) respectively.

These laws are essential for transforming complex expressions into simpler forms, making calculations straightforward and error-free.