Problem 99 - Elementary Algebra

Question

Simplify \(\left(x^{3} y^{2} z^{5}\right)^{6}\left(x^{2} y z\right)^{2}\)

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(x^{3} y^{2} z^{5}\right)^{6} *\left(x^{2} y z\right)^{2}\). Answer: \(x^{22}y^{14}z^{32}\)

Step 1: Apply the power of a power law to each term within each parentheses.

Using the power of a power law for \(\left(x^{3} y^{2} z^{5}\right)^{6}\) and \(\left(x^{2} y z\right)^{2}\): \((x^3)^6(x^2)^2 = x^{3*6}y^{2*6}z^{5*6}x^{2*2}y^{1*2}z^{1*2}\)

Step 2: Simplify the exponents

Simplify the exponents for each term: \(x^{18}y^{12}z^{30}x^{4}y^{2}z^{2}\)

Step 3: Apply the product of powers law to combine the terms with the same base

Using the product of powers law to combine the terms: \(x^{18+4}y^{12+2}z^{30+2} = x^{22}y^{14}z^{32}\) So, the simplified expression is \(x^{22}y^{14}z^{32}\).

Key Concepts

Power of a Power Law

The power of a power law is a fundamental rule in exponents that helps simplify expressions where an exponent is raised to another exponent. It states that when you have an expression \((a^m)^n\), you can multiply the exponents: \[a^{m \cdot n}\]. This is particularly useful in breaking down complex expressions into simpler forms.
Let's take the expression from our original problem: \((x^3 y^2 z^5)^6\). Applying the power of a power law means that each exponent in the original term inside the parenthesis is multiplied by the outside exponent, 6, resulting in:

• For \(x^3\), it becomes \(x^{3 \cdot 6} = x^{18}\).
• For \(y^2\), it becomes \(y^{2 \cdot 6} = y^{12}\).
• For \(z^5\), it becomes \(z^{5 \cdot 6} = z^{30}\).

This application allows us to significantly simplify the expression and make it more manageable.
Remember, the power of a power doesn't add the exponents; it multiplies them. This distinction is crucial for accurate calculations.

Product of Powers Law

In mathematics, when you multiply two powers with the same base, the exponents are added together. This rule is known as the product of powers law, and it's essential for combining terms efficiently. The general formula for this law is: \(a^m \cdot a^n = a^{m+n}\).
Applying it to the exercise, we take the expressions we obtained after applying the power of a power law:

• \(x^{18}\) and \(x^4\)
• \(y^{12}\) and \(y^2\)
• \(z^{30}\) and \(z^2\)

For the base \(x\), you add the exponents:\(x^{18} \cdot x^4 = x^{18+4} = x^{22}\).
Similarly, for \(y\) and \(z\):

• \(y^{12} \cdot y^2 = y^{14}\)
• \(z^{30} \cdot z^2 = z^{32}\)

This approach dramatically simplifies the final expression, making it \(x^{22}y^{14}z^{32}\). By understanding and applying the product of powers law, you create cleaner and more elegant algebraic expressions.

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. They comprise a base and a power (also called an exponent). For instance, in \(x^a\), \(x\) is the base, and \(a\) is the exponent. The term represents the base being multiplied by itself \(a\) times.
Exponents are fundamental in mathematics because they allow us to write large numbers more compactly and to perform extensive calculations more easily.
There are several rules to follow when working with exponents:

• **Multiplication:** Multiplying like bases with exponents adds the exponents.
• **Division:** Dividing like bases with exponents subtracts the exponents.
• **Power of a power:** Raising a power to another exponent multiplies the exponents.
• **Zero exponent rule:** Any base with an exponent of zero equals 1, for example, \(x^0 = 1\).

Understanding these rules ensures that you can simplify expressions accurately and solve problems efficiently. Familiarity with exponents is crucial not just in mathematics, but also in fields like science and engineering, where large numbers and complex calculations are common.