Problem 57 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \frac{\left(10 x^{4} y^{5} z^{11}\right)^{3}}{\left(x y^{2}\right)^{4}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(\frac{\left(10 x^{4} y^{5} z^{11}\right)^{3}}{\left(x y^{2}\right)^{4}}\) Answer: \(1000x^8y^7z^{33}\)

Step 1: Apply the power of a product rule

The power of a product rule states that \((ab)^{n} = a^{n}b^{n}\). In our exercise, we are given: $$ \frac{\left(10 x^{4} y^{5} z^{11}\right)^{3}}{\left(x y^{2}\right)^{4}} $$ Applying the power of a product rule, we have: $$ \frac{10^{3}x^{4\cdot 3}y^{5\cdot 3}z^{11\cdot 3}}{x^{1\cdot 4}y^{2\cdot 4}} $$

Step 2: Simplify the exponents

Now, we will simplify the exponents by multiplying them: $$ \frac{10^3 x^{12} y^{15} z^{33}}{x^4 y^8} $$

Step 3: Apply the division rule for exponents

The division rule for exponents states that \(\dfrac{a^{m}}{a^{n}} = a^{m-n}\). So, we can apply this rule to our expression to combine the powers of x and y: $$ 10^3 x^{12-4} y^{15-8} z^{33} $$

Step 4: Simplify the final expression

Finally, we will simplify the final expression by calculating the exponents: $$ 1000 x^8 y^7 z^{33} $$ So, the simplified expression is: \(1000x^8y^7z^{33}\).

Key Concepts

Power of a Product Rule

When you see a term with multiple factors raised to an exponent, don't worry, it's simpler than it looks! The "power of a product rule" says that we can deal with each factor separately. If we have something like \((ab)^n\), this breaks down into \(a^n \cdot b^n\). It's similar to distributing the exponent across every part of the product inside the parentheses.

In our exercise, we started with \((10x^4y^5z^{11})^3\). This means we apply the exponent 3 to each element inside the parentheses:

• \(10^3\)
• \(x^{4 \cdot 3} = x^{12}\)
• \(y^{5 \cdot 3} = y^{15}\)
• \(z^{11 \cdot 3} = z^{33}\)

By doing this, you simplify complex expressions into manageable pieces.

Division Rule for Exponents

Dividing expressions with exponents is a handy skill to have. The "division rule for exponents" tells us that if we have the same base being divided, we can simplify it by subtracting the exponents. It looks like this: \(\frac{a^m}{a^n} = a^{m-n}\).

This is exactly what we did in the provided expression after simplifying the powers. For \(\frac{10^3 x^{12} y^{15} z^{33}}{x^4 y^8}\), we take:

• \(x^{12-4} = x^8\), because you subtract 4 from 12.
• \(y^{15-8} = y^7\), subtracting 8 from 15.

The key is consistency—only apply this to terms with the same base. Objects like \(z^{33}\) remain unchanged if there is no corresponding \(z\) in the denominator.

Simplifying Expressions

Once we've applied both the power of a product rule and the division rule for exponents, what remains is to simplify what's left. Often this is just combining or reducing the final terms into a single statement that is easy to understand. In our current example, the simplification involved obtaining the final expression of \(1000 x^8 y^7 z^{33}\).

The number \(10^3\) becomes \(1000\), as we compute \(10 \times 10 \times 10\). The variables \(x\), \(y\), and \(z\) follow naturally from applying and simplifying their respective exponents.

In essence, the simplification process is about tidying up the work you’ve done using the rules. It’s like clearing off the clutter to see a clean, finished result. Each part of these expressions behaves consistently with the rules, allowing you to confidently simplify complex expressions into more digestible forms.