Problem 111 - College Algebra

Question

Simplify each exponential expression.Assume that variables represent nonzero real numbers. $$\left(\frac{x^{3} y^{4} z^{5}}{x^{-3} y^{-4} z^{-5}}\right)^{-2}$$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \( 1/(x^{12} y^{16} z^{20}) \)

Step 1: Simplify The Inside Expression

Before dealing with the power outside the fraction, let's simplify the fractions inside the bracket using the rule for dividing by an exponent: \( x^{3}/x^{-3} = x^{3-(-3)} = x^{6} \), \( y^{4}/y^{-4} = y^{4-(-4)} = y^{8} \), and \( z^{5}/z^{-5} = z^{5-(-5)} = z^{10} \). Therefore, the inside of the bracket becomes: \( (x^{6} y^{8} z^{10}) \)

Step 2: Apply The Power Of A Power Rule

Now that the inside of the bracket is simplified, next apply the power of a power rule to the -2 power outside the bracket to each term inside the bracket: \( (x^{6})^{-2} = x^{(6*-2)} = x^{-12} \), \( (y^{8})^{-2} = y^{(8*-2)} = y^{-16} \), \( (z^{10})^{-2} = z^{(10*-2)} = z^{-20} \)

Step 3: Apply The Negative Exponent Rule

Once the power of a power rule is applied, apply the negative exponent rule to each of the variables to further simplify: \( x^{-12} = 1/x^{12} \), \( y^{-16} = 1/y^{16} \), \( z^{-20} = 1/z^{20} \)

Step 4: Combine The Terms

Then, combine all the terms into a singular expression: \( 1/(x^{12} y^{16} z^{20}) \)

Key Concepts

Power of a Power Rule

Understanding the power of a power rule is essential for dealing with expressions that have exponents. Imagine you have an expression like

• \((a^m)^n\)

The power of a power rule states that you multiply the exponents. So, you end up with

• \(a^{m \cdot n}\)

This is particularly helpful when you're looking at expressions within parentheses that are raised to another power. In our original exercise, after simplifying the inside of the fraction, we applied the \((-2)\) power to each term:

• \((x^6)^{-2} = x^{-12}\)
• \((y^8)^{-2} = y^{-16}\)
• \((z^{10})^{-2} = z^{-20}\)

This rule saves time and simplifies your work by neatly multiplying the exponents.

Negative Exponent Rule

The negative exponent rule may seem tricky at first, but it's actually a simple way to handle negative powers. The rule says:

• \(a^{-n} = \frac{1}{a^n}\)

A negative exponent indicates that you should take the reciprocal of the base raised to the positive of that exponent. In the exercise, after applying the power of a power rule, each term had a negative exponent:

• \(x^{-12} = \frac{1}{x^{12}}\)
• \(y^{-16} = \frac{1}{y^{16}}\)
• \(z^{-20} = \frac{1}{z^{20}}\)

By converting negative exponents into positive ones, we make expressions easier to interpret and work with.

Simplifying Expressions

Simplifying expressions involves combining all operations into their simplest form. This makes expressions easier to understand and work with. Let's break down the process:

• First, simplify each part of the expression by addressing the operations inside parentheses.
• Then, handle all exponents using rules like the power of a power rule and the negative exponent rule.
• Finally, combine all parts into a simplified, single expression.

In our exercise, after breaking down the exponents and handling negative exponents, the expression was fully simplified to:

• \(\frac{1}{x^{12} y^{16} z^{20}}\)

This final expression represents a tidied up version of the original, expressing the value clearly and concisely.