Problem 55 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponents \((\mathrm{s}) .\) Assume that all letters denote positive numbers. \(\left(\frac{x^{6} y}{y^{4}}\right)^{5 / 2}\)

Step-by-Step Solution

Answer

\( \frac{x^{15}}{y^{\frac{15}{2}}} \).

Step 1: Simplify the Fraction

Start by simplifying the fraction inside the parentheses. We have \( \frac{x^{6} y}{y^{4}} \). This can be simplified by subtracting the exponents of \(y\): \[ \frac{x^{6} y}{y^{4}} = x^6 y^{1-4} = x^6 y^{-3}. \] So, the expression simplifies to \( x^6 y^{-3} \).

Step 2: Apply the Power to Each Factor

Next, we need to apply the exponent \( \frac{5}{2} \) to each part of the simplified expression \( x^6 y^{-3} \). Using the power of a power property \((a^m)^n = a^{m\cdot n}\), we have: \[ (x^6 y^{-3})^{\frac{5}{2}} = (x^6)^{\frac{5}{2}}(y^{-3})^{\frac{5}{2}}. \]

Step 3: Simplify the Powers of X and Y

Now calculate each part separately: \[ (x^6)^{\frac{5}{2}} = x^{6 \times \frac{5}{2}} = x^{15}, \] and \[ (y^{-3})^{\frac{5}{2}} = y^{-3 \times \frac{5}{2}} = y^{-\frac{15}{2}}. \]

Step 4: Eliminate Negative Exponents

Finally, eliminate the negative exponent by rewriting \( y^{-\frac{15}{2}} \) as \( \frac{1}{y^{\frac{15}{2}}} \). The expression becomes: \[ x^{15} y^{-\frac{15}{2}} = \frac{x^{15}}{y^{\frac{15}{2}}}. \]

Key Concepts

Exponent Rules

Exponent rules form the foundation of operations involving powers and indices. They help us simplify expressions, especially when dealing with variables to different powers. Here are some key rules:

• Product of Powers Rule: When multiplying two expressions with the same base, add the exponents: \( a^m imes a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing two expressions with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \). This rule was applied when simplifying \( \frac{x^6 y}{y^4} \) in the problem.
• Zero Exponent Rule: Any base raised to the zero power is always 1: \( a^0 = 1 \), assuming \( a eq 0 \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n} \). This principle is used in Step 2 to simplify \((x^6 y^{-3})^{\frac{5}{2}}\).

Understanding these rules allows you to manipulate and simplify complex expressions effectively. They are particularly useful when dealing with algebraic formulas.

Negative Exponents

Negative exponents can sometimes seem challenging, but they are actually quite straightforward once you grasp the underlying concept. A negative exponent simply means the reciprocal of the base raised to the opposite positive exponent.
For example, \( a^{-m} = \frac{1}{a^m} \).

• This means that if you have a negative exponent, you can rewrite it as the reciprocal of the positive exponent form, just as we did with \( y^{-\frac{15}{2}} = \frac{1}{y^{\frac{15}{2}}} \) in the solution.
• Negative exponents effectively "move" the base to the other part of the fraction, like from the numerator to the denominator or vice versa.

By transforming negative exponents into fractions, we make expressions easier to interpret and work with, especially when simplifying or solving algebraic equations. This concept emphasizes that an exponent tells you how many times to multiply the base, while a negative sign denotes structure in terms of division.

Power of a Power

The "power of a power" rule is one of the essential exponent laws that can simplify expressions significantly. This rule states that raising an exponentiated term to another power is equivalent to multiplying the exponents.
For example, \((a^m)^n = a^{m \cdot n}\).

• This is extremely useful in simplifying nested or composite exponents -- exponents of exponents, like in the problem's \((x^6 y^{-3})^{\frac{5}{2}}\).
• Applying this rule first ensures all powers are simplified correctly before tackling negative exponents or implementing other exponent rules.

The practical application of this rule involves multiplying the outside exponent by each exponent inside the parentheses. This multiplication leads to simpler expressions that are easier to further manipulate or resolve. Additionally, this rule helps in unraveling complex terms into more elementary forms, making solving algebraic equations more approachable.