Problem 62 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(\frac{x^{4} z^{2}}{4 y^{5}}\right)\left(\frac{2 x^{3} y^{2}}{z^{3}}\right)^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Second Fraction

Start by simplifying the fraction inside the parentheses on the right. Notice the expression \( \left(\frac{2x^{3}y^{2}}{z^{3}}\right)^{2}\).Apply the power of a quotient rule: \(\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}\), yielding:\[ \frac{(2)^{2} \cdot (x^{3})^{2} \cdot (y^{2})^{2}}{(z^{3})^{2}} \]Which simplifies to:\[ \frac{4x^{6}y^{4}}{z^{6}} \]

Step 2: Multiply the Expressions

Now multiply the two fractions:\[\frac{x^{4}z^{2}}{4y^{5}} \cdot \frac{4x^{6}y^{4}}{z^{6}} \]Use the property of multiplication of fractions:\(\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\)Resulting in:\[\frac{x^{4}z^{2} \cdot 4x^{6}y^{4}}{4y^{5}z^{6}}\]

Step 3: Simplify the Expression

Perform the multiplication:- Numerator: \(x^{4} \cdot x^{6} = x^{10}\), - Combine the constants: \(z^{2} \cdot 4\) gives \(4z^{2}\times 4 = 4z^{2}\), - Combine powers of \(y\): \(y^{4}\)Resulting in:\[\frac{4x^{10}y^{4}z^{2}}{4y^{5}z^{6}}\]

Step 4: Simplify the Fraction

Cancel common factors in the numerator and the denominator:- The 4s cancel out: \(\frac{4}{4} = 1\).- Divide \(y\) terms: \(\frac{y^{4}}{y^{5}} = \frac{1}{y}\).- Divide \(z\) terms: \(\frac{z^{2}}{z^{6}} = \frac{1}{z^{4}}\).The simplified expression is:\[x^{10}\cdot\frac{1}{yz^{4}}\]

Key Concepts

Power of a Quotient Rule

The power of a quotient rule is a fundamental concept in algebra that helps in simplifying expressions where a fraction is raised to a power. This rule states:

• \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

It implies that when a fraction \(\frac{a}{b}\) is raised to the power \(n\), you apply the exponent \(n\) to both the numerator \(a\) and the denominator \(b\) separately. This simplifies more complex expressions effectively.

In our exercise, this rule was applied to simplify

• \(\left(\frac{2x^{3}y^{2}}{z^{3}}\right)^2\)

By applying the power of a quotient rule, it becomes:

• \(\frac{2^2 \cdot (x^3)^2 \cdot (y^2)^2}{(z^3)^2} = \frac{4x^6y^4}{z^6}\)

This is a crucial step to handle exponents efficiently while working through more complex algebraic manipulations.

Multiplying Fractions

Multiplying fractions involves a straightforward process of multiplying the numerators together and the denominators together. The general formula is:

• \(\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\)

This means the top numbers (numerators) are multiplied with each other and the bottom numbers (denominators) are multiplied with each other.

In our exercise, we have two fractions:

• \(\frac{x^4z^2}{4y^5}\)
• \(\frac{4x^6y^4}{z^6}\)

Multiplying them, according to the formula, results in:

• \(\frac{x^4z^2 \cdot 4x^6y^4}{4y^5z^6}\)

This simplification makes solving algebraic expressions much faster, especially when dealing with variables and exponents.

Negative Exponents

Negative exponents are another key algebraic concept. A negative exponent indicates that a number should be taken as a reciprocal. In simple terms:

• \(a^{-n} = \frac{1}{a^n}\)

This is particularly useful in simplifying expressions where the exponents might appear confusing.

In terms of our exercise, eliminating any negative exponents is critical to simplifying the final result. Whenever you encounter a negative exponent in an expression, it's important to flip the base to the reciprocal position. During the last steps of simplifying, negative exponents, if any, are strategically moved and expressed as reciprocals to ensure clarity and simplicity in final results. Thus, negative exponents are transformed to positive by putting them in the denominator.

Algebraic Manipulation

Algebraic manipulation allows us to make changes to an expression in order to simplify it, solve an equation, or to explore a mathematical model. This involves strategies such as factoring, expanding, and simplifying numbers and variables.

In our exercise, algebraic manipulation plays a pivotal role. Simplifying the expression \(\frac{4x^{10}y^{4}z^{2}}{4y^{5}z^{6}}\) involves:

• Cancelling out the constants: \(\frac{4}{4} = 1\).
• Simplifying the terms of \(y\): \(\frac{y^{4}}{y^{5}} = \frac{1}{y}\).
• Simplifying \(z\) terms: \(\frac{z^{2}}{z^{6}} = \frac{1}{z^{4}}\).

Through these steps of algebraic manipulation, the expression is reduced to its simplest form: \(x^{10} \cdot \frac{1}{yz^4}\). Breaking down each part facilitates understanding and executing similar problems, especially in mathematics where clarity and accuracy are essential.