Problem 144 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. Simplify \(\left(4 x^{5} y^{3} z^{0}\right)^{3}\)

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(4 x^{5} y^{3} z^{0}\right)^{3}\) and ensure only positive exponents appear. Answer: \(64x^{15}y^9\)

Step 1: Apply the power-rule of exponents

The expression that we want to simplify is \(\left(4 x^{5} y^{3} z^{0}\right)^{3}\). To do this, raise each term within the parentheses to the power of 3, using the power rule of exponents: \((a b)^n = a^n b^n\). We get \((4^3)(x^{5\cdot3})(y^{3\cdot3})(z^{0\cdot3})\).

Step 2: Evaluate the exponents

Calculate the result of each exponent: - \(4^3 = 64\) - \(x^{5\cdot3} = x^{15}\) - \(y^{3\cdot3} = y^9\) - \(z^{0\cdot3} = z^0\)

Step 3: Simplify the expression

Combine the results from the previous step into a simplified expression: \(64x^{15}y^9z^0\). Since any variable or number to the power of 0 is 1 by definition, we can further simplify the expression by removing \(z^0\) and replacing it with 1: \(64x^{15}y^9(1)\).

Step 4: Final simplification

The expression \(64x^{15}y^9(1)\) is the same as \(64x^{15}y^9\), because any number multiplied by 1 remains unchanged. So, the simplified expression with only positive exponents is \(\boxed{64x^{15}y^9}\).

Key Concepts

Power Rule in Exponents

The power rule is an essential component when working with exponents. It allows us to simplify expressions raised to a power efficiently. When you have a product within a parenthesis, like \((ab)^n\), you can distribute the exponent \(n\) across each factor inside the parenthesis. This means the expression \((ab)^n\) becomes \(a^n b^n\).
For instance, in the expression \((4x^5y^3z^0)^3\), we apply the power rule to get \((4^3)(x^{5\cdot3})(y^{3\cdot3})(z^{0\cdot3})\).

• The number and each variable are raised to the power of 3.
• The rule simplifies complex expressions, making it manageable to solve them.

The power rule is a simple yet powerful tool that makes calculations involving exponents straightforward and error-free.

Understanding Exponents

Exponents represent repeated multiplication and are a fundamental part of algebra. An expression such as \(x^5\) means "\(x\) multiplied by itself 5 times."
Exponents follow certain rules:

• Anything to the power of 0 is 1 (\(z^0 = 1\)).
• Multiplying powers with the same base involves adding the exponents (\(x^m \cdot x^n = x^{m+n}\)).
• Raising a power to another power means multiplying the exponents (\((x^m)^n = x^{m\cdot n}\)).

These rules help simplify expressions and understand the behavior of numbers and variables under multiplication.

Simplifying Expressions

Simplifying expressions is the process of reducing an expression to its most concise form while maintaining its value. It often involves several steps:
In our problem,

• First, the expression \((4x^5y^3z^0)^3\) was expanded using the power rule resulting in \((4^3)(x^{15})(y^9)(z^0)\).
• Then, each component was calculated: \(4^3 = 64\), \(x^{15}\), \(y^9\), and \(z^0 = 1\).
• Subsequently, any factor equaled to 1, like \(z^0\), can be removed from the expression.
• Finally, the expression becomes \(64x^{15}y^9\).

Simplifying expressions reduces complexity, making them easier to work with in further calculations or applications.