Problem 64 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \frac{5 x y^{-2}}{x^{-1} y^{-3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(5x^2y\).

Step 1: Apply the Negative Exponent Rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). Let's first rewrite each negative exponent in the expression to eliminate them:In the numerator: - The term \(y^{-2}\) becomes \(\frac{1}{y^2}\).In the denominator: - The term \(x^{-1}\) becomes \(\frac{1}{x}\).- The term \(y^{-3}\) becomes \(\frac{1}{y^3}\).Substituting these back, we get:\[ \frac{5x \cdot \frac{1}{y^2}}{\frac{1}{x} \cdot \frac{1}{y^3}} \]

Step 2: Simplify the Fraction

We can now combine the numerators and denominators:\[ \frac{5x \cdot \frac{1}{y^2}}{\frac{1}{x} \cdot \frac{1}{y^3}} = \frac{5x}{y^2} \div \frac{1}{xy^3} \]Dividing by a fraction is the same as multiplying by its reciprocal. Thus, we rewrite:\[ \frac{5x}{y^2} \times \frac{xy^3}{1} \]

Step 3: Simplify the Multiplication

Multiply the fractions:- Multiply the numerators: \(5x \cdot x = 5x^2\)- Multiply the denominators: \(y^2 \cdot 1 = y^2\)Now the expression is:\[ \frac{5x^2 \cdot y^3}{y^2} \]

Step 4: Simplify the Expression Further

When simplifying \(\frac{y^3}{y^2}\), subtract the exponents of like bases according to \(a^m / a^n = a^{m-n}\). This gives:\(y^{3-2} = y^1 = y\)Substitute back:\[ \frac{5x^2 \cdot y}{1} = 5x^2y \]

Step 5: Present the Final Answer

The given expression simplifies to:\(5x^2y\)

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing an expression to its simplest form, making it easier to work with. In this exercise, the goal is to eliminate negative exponents and simplify the fractional expression. Simplifying helps identify and remove any unnecessary complexity in the expression.

Here's how you can approach simplifying algebraic expressions:

• Identify terms with negative exponents and apply the negative exponent rule.
• Rewrite these terms as positive exponents by flipping the fraction (Invert the base).
• Simplify any fractions by multiplying or dividing as necessary.
• Combine like terms, using basic arithmetic operations and exponent rules.

Reducing an expression to its simplest form often reveals insights about the relationships between variables and can make further calculations or problem-solving more straightforward.

Exponent Rules in Algebra

Exponent rules are crucial for manipulating and simplifying expressions that involve powers. There are several key exponent rules that come in handy, especially in solving problems like this one.

The exercise highlights the following exponent rule:

• Negative Exponent Rule: This rule provides that any number raised to a negative exponent is equivalent to the reciprocal of that number raised to the opposite positive exponent. Mathematically, this is expressed as \(a^{-n} = \frac{1}{a^n}\).

Additionally, you'll need to remember:

• Product of Powers Rule: This states that when multiplying two powers with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing two powers with the same base, you subtract the exponents: \(a^m / a^n = a^{m-n}\).

The application of these rules allows the expression to be simplified effectively, breaking down complex terms into manageable calculations.

Fractional Exponents Simplification

Simplifying expressions with fractional exponents requires understanding how these exponents work. Fractional exponents express roots and can be a bit tricky if you're new to them.

Key points about fractional exponents:

• The numerator of a fractional exponent signifies the power, while the denominator indicates the root.
• For instance, \(a^{\frac{m}{n}}\) can be interpreted as \(\sqrt[n]{a^m}\) or \((\sqrt[n]{a})^m\).

In the context of the given problem, while there are no fractional exponents used explicitly, understanding this concept can be helpful in manipulating expressions where these appear, especially when dealing with both positive and negative exponents.

Simplifying fractional exponents often involves rewriting them in the simplest radical form and applying other exponent rules for further simplification. Managing fractional exponents with these techniques ensures expressions are reduced to their best form, easing the complexity further down in calculations.