Problem 123 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression.$$\left(64 x^{3} y^{4}\right)^{-3}\left(8 x^{3} y^{2}\right)^{4}$$

Step-by-Step Solution

Answer

\(0.015625 * x^{3}/y^{4}\)

Step 1: Apply the power of a power rule

The expression \(\left(64 x^{3} y^{4}\right)^{-3}\left(8 x^{3} y^{2}\right)^{4}\) can be rewritten by applying the power of a power rule, which states that \((a^m)^n = a^{mn}\), where \(a\), \(m\), and \(n\) are real numbers. We get \((64^{-3} x^{-9} y^{-12})(8^{4} x^{12} y^{8})\).

Step 2: Simplify the bases and the exponents

Now, we need to simplify the expression by performing the operations on the bases and the exponents. We have \(64^{-3} = 1/64^3 = 1/262144\) and \(8^4 = 8 * 8 * 8 * 8 = 4096\). Thus, the expression becomes \((1/262144 * x^{-9} * y^{-12})(4096 * x^{12} * y^{8})\).

Step 3: Perform multiplication

Now, perform multiplication to simplify the expression. We get \(1/262144 * 4096 * x^{(-9+12)} * y^{(-12+8)}\).

Step 4: Final simplification

Further simplifying, we obtain \(4096/262144 * x^3 * y^{-4}\) which equals \(x^3 / y^4 * 0.015625\).

Key Concepts

Power of a Power Rule

The power of a power rule is an essential exponent rule in algebra. When you see an expression like \((a^m)^n\), you can simplify it to \(a^{mn}\). This rule helps simplify complex terms in an expression by multiplying the exponents.

In the original exercise, we applied this rule to simplify \( (64 x^{3} y^{4})^{-3} \) and \( (8 x^{3} y^{2})^{4} \).

• For \((64 x^{3} y^{4})^{-3}\), it becomes \(64^{-3} x^{-9} y^{-12}\).
• For \((8 x^{3} y^{2})^{4}\), it becomes \(8^{4} x^{12} y^{8}\).

By applying the power of a power rule, we transform nested exponents into simpler expressions, making it easier to work with the terms in subsequent steps.

Exponent Rules

Exponent rules are key tools when working with powers and simplify expressions efficiently. These rules include ways to multiply and divide powers, as well as handle negative exponents.

In the exercise, the following exponent rules were particularly useful:

• Negative Exponents: Any base with a negative exponent can be rewritten as its reciprocal with a positive exponent: \(a^{-m} = \frac{1}{a^m}\).
• Muliplying Exponents with the Same Base: Add the exponents together: \((x^m)(x^n) = x^{m+n}\).

We used these rules to simplify exponents and bases throughout the process, transforming the expression \((64^{-3} x^{-9} y^{-12})(8^{4} x^{12} y^{8})\) into more manageable pieces.

Multiplication of Expressions

The multiplication of expressions combines individual terms to create a new, simplified expression. We utilize multiplication properties and exponent rules to achieve this.

In the exercise, after rewriting the expression using the power of a power rule, we multiplied the terms:

• The constants \(1/262144\) and \(4096\) were multiplied to give the simplified constant.
• The terms with the variable \(x\) used the addition of exponents \((-9 + 12)\) resulting in \(x^3\).
• For \(y\), we added exponents \((-12+8)\) obtaining \(y^{-4}\), which is then simplified further.

This method reduces a complex product into a single expression \(x^3 / y^4 * 0.015625\), making it much easier to understand and work with.