Problem 56 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(\frac{y^{3}}{x^{-4}}\right)^{-2}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{x^8}{y^6}\).

Step 1: Simplify Fraction Exponents

The given expression is \(\left(\frac{y^3}{x^{-4}}\right)^{-2}\). First, realize that any term raised to a negative exponent means you take the reciprocal. Therefore, \(x^{-4}\) becomes \(x^{4}\) when you move it to the numerator. The expression inside the parentheses changes to \(\frac{y^3 \cdot x^4}{1}\).

Step 2: Apply Negative Exponent

Now, apply the negative exponent to the entire fraction. Since we have \(\left(\frac{y^3 \cdot x^4}{1}\right)^{-2}\), apply \(-2\) to both \(y^3\) and \(x^4\). This gives us \(\frac{1}{y^6 \cdot x^8}\) because \((y^3)^{-2} = y^{-6}\) and \((x^4)^{-2} = x^{-8}\), which become positive exponents when inverted.

Step 3: Write the Final Expression

Finally, the expression simplifies to \(\frac{x^8}{y^6}\). This is because the negative exponents have been moved across the fraction line, turning them positive.

Key Concepts

Negative Exponents

Negative exponents may look intimidating, but they're quite simple once you understand the core idea. A negative exponent indicates that we must take the reciprocal of the base and then apply the positive version of the exponent. For instance, if you have a term like \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\). This rule applies uniformly across different terms.
In the original exercise, the term \(x^{-4}\) is transformed by moving \(x\) to the numerator, which makes it \(x^4\). Using this rule:

• The negative sign indicates the term should "flip" to the opposite part of the fraction.
• Once flipped, the exponent becomes positive.

This transformation ensures all expressions are simplified to only involve positive exponents, making subsequent calculations more straightforward.

Fractional Exponents

Fractional exponents may seem complex but are quite understandable with a little breakdown. A fractional exponent indicates both a root and a power. For example, an expression like \(a^{\frac{m}{n}}\) is equal to the \(n\)-th root of \(a\), all raised to the power of \(m\). Depending on how you approach it, you can perform these operations in either order:

• First take the \(n\)-th root of \(a\), then raise it to the \(m\)-th power.
• Or, first raise \(a\) to the \(m\)-th power, then take the \(n\)-th root.

While this exercise doesn't directly involve fractional exponents, it is important to understand their relationship with radical expressions. Knowing how they work will prepare you for scenarios where both exponents and roots are involved.

Simplifying Expressions

Simplifying expressions is much like tidying up a room; you want everything in its proper place. The goal is to make equations as neat and manageable as possible, often by eliminating negative exponents or complex fractions.

• Start by addressing negative exponents: change them to positive by flipping the term's position in the fraction.
• Ensure all expressions are expressed with positive exponents for clarity and simplicity.
• Combine like terms and reduce any fractions when applicable.

In the provided exercise, the expression \(\left(\frac{y^3}{x^{-4}}\right)^{-2}\) is simplified by first addressing the negative exponent on \(x\), then applying the outer negative exponent \(-2\). This process reduces the expression to \(\frac{x^8}{y^6}\), demonstrating how consistently applying exponent rules can simplify seemingly complex expressions.