Problem 54 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(4 x^{5} y^{-2}\right)^{-2}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{y^4}{16x^{10}}\).

Step 1: Apply the Power of a Power Rule

The expression given is \(\left(4x^5y^{-2}\right)^{-2}\). Apply the power of a power rule, \((a^m)^n = a^{m\cdot n}\), to each term inside the parentheses. This gives:\[4^{-2}\cdot (x^5)^{-2}\cdot (y^{-2})^{-2}\]

Step 2: Simplify Each Component

Now, simplify each component from the expression using the multiplication of exponents with the power of a power rule:- For \(4^{-2}\), it remains \(4^{-2}\).- For \((x^5)^{-2}\), multiply the exponents: \(x^{5 \cdot (-2)} = x^{-10}\).- For \((y^{-2})^{-2}\), multiply the exponents: \(y^{-2 \cdot (-2)} = y^4\).So the expression simplifies to:\[4^{-2}\cdot x^{-10}\cdot y^4\]

Step 3: Convert Negative Exponents to Positive

Use the definition of negative exponents: \(a^{-n} = \frac{1}{a^n}\) to convert the expression with negative exponents into positive ones:\[ = \frac{y^4}{4^2 \cdot x^{10}}\]

Step 4: Simplify Final Expression

Calculate \(4^2\) and substitute it into the fraction. We have:\(4^2 = 16\), so the expression becomes:\[\frac{y^4}{16x^{10}}\]

Key Concepts

Exponents

Exponents are fundamental building blocks of algebra that express repeated multiplication of a number by itself. In the expression, the exponent is the small number positioned above and to the right of the base number. For instance, in \(4^2\), 4 is the base, and 2 is the exponent. This means \(4\) is to be multiplied by itself, resulting in \(4 \times 4\) or 16.

Exponents have certain rules that help simplify mathematical expressions, such as:

• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)

Understanding these rules is crucial for algebraic simplification because they provide methods to combine or break down expressions efficiently.

In the given exercise, recognizing when and how to apply these rules is key to reaching the simplified final form.

Negative Exponents

Negative exponents can be a bit tricky, but they serve an important role in simplifying expressions by moving terms from the numerator to the denominator and vice versa. The expression \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\). This means that, rather than interpreting negative exponents as negative numbers, they should be viewed as a direction for moving the base from its current position.

Here’s how this applies to the given exercise:

• \(4^{-2}\) becomes \(\frac{1}{4^2}\), and calculating further gives \(\frac{1}{16}\)
• \(x^{-10}\) becomes \(\frac{1}{x^{10}}\)

Remembering this can help greatly in effectively simplifying expressions, reducing potential confusion over negative follow-on computations.

Power of a Power Rule

The power of a power rule combines exponents in expressions where an exponent is raised to another exponent. This rule is represented as \((a^m)^n = a^{m \cdot n}\). This rule allows us to condense expressions by directly multiplying the exponents. It's the interplay of these powers that can take a confusing expression and change it into a single, more manageable form.

Let's illustrate how this works in the provided problem:

• For \((x^5)^{-2}\), we use: \(x^{5 \times (-2)} = x^{-10}\)
• For \((y^{-2})^{-2}\), we use: \(y^{-2 \times (-2)} = y^4\)

This step simplifies the expression considerably, paving the way for transforming negative exponents into their positive counterparts, enhancing the overall simplification process.

Algebraic Expression

An algebraic expression is a mathematical phrase that can comprise numbers, variables, and operation symbols. These expressions might seem complex, but they can be broken down and simplified using algebraic rules and properties. The goal is to make them easier to evaluate or understand.

The initial algebraic expression in our task \((4x^5y^{-2})^{-2}\), uses all these elements. Variables like \(x\) and \(y\), coefficients like \(4\), and exponents are put together within parentheses. Simplifying such expressions involves strategically applying rules:

• The power of a power rule to handle terms within parentheses.
• The transformation of negative exponents to their positive forms.

Through our step-by-step approach, each layer is unwrapped to reveal a much more digestible form: \(\frac{y^4}{16x^{10}}\). Understanding each component of an algebraic expression and knowing which rules to employ provides clarity and efficiency in solving similar algebraic problems.