Problem 66 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ \left(\frac{4 y^{3} z^{2 / 3}}{x^{1 / 2}}\right)^{2}\left(\frac{x^{-3} y^{6}}{8 z^{4}}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{8y^8}{x^2} \).

Step 1: Expand and Simplify Each Term Separately

For the first term, use the power rule: \[\left(\frac{4y^3z^{2/3}}{x^{1/2}}\right)^2 = \frac{4^2(y^3)^2(z^{2/3})^2}{(x^{1/2})^2} = \frac{16y^6z^{4/3}}{x}\]For the second term, use the power rule:\[\left(\frac{x^{-3}y^6}{8z^4}\right)^{1/3} = \frac{(x^{-3})^{1/3}(y^6)^{1/3}}{(8z^4)^{1/3}} = \frac{x^{-1}y^2}{2z^{4/3}}\]

Step 2: Combine the Expanded and Simplified Terms

Multiply the two results obtained in Step 1:\[\left(\frac{16y^6z^{4/3}}{x}\right) \times \left(\frac{x^{-1}y^2}{2z^{4/3}}\right) = \frac{16y^6z^{4/3} \cdot x^{-1}y^2}{x \cdot 2z^{4/3}} = \frac{16x^{-1}y^8z^{4/3}}{2xz^{4/3}} \]

Step 3: Simplify the Resulting Expression

Cancel out common terms in the numerator and denominator:\[\frac{16x^{-1}y^8z^{4/3}}{2xz^{4/3}} = \frac{16y^8}{2x^2} = 8\frac{y^8}{x^2}\]Since all letters denote positive numbers, the expression is simplified without negative exponents.

Key Concepts

Negative Exponents

In algebra, negative exponents can be tricky, but they are easy to understand once you get the hang of them. Simply put, a negative exponent indicates the reciprocal of a base raised to the corresponding positive exponent.
For example, if you have an expression like \( x^{-3} \), it translates to \( \frac{1}{x^3} \). The negative sign in the exponent tells you to "flip" the base to the denominator.

• Convert \( a^{-n} \) to \( \frac{1}{a^n} \).
• This helps in handling expressions and simplifying them when negative exponents are involved.

Understanding this allows you to manipulate and simplify algebraic expressions efficiently. By removing negative exponents, the resulting expressions are often easier to work with and interpret.

Power Rule

The power rule is a key concept when working with exponents and algebraic expressions. It states that when you raise a power to another power, you multiply the exponents.
For example, \( (a^m)^n \) becomes \( a^{m \times n} \). This rule is handy when simplifying complex expressions or expanding terms within parentheses.

• Apply \( (x^m)^n = x^{m \times n} \).
• This makes calculations more straightforward and the expressions cleaner.
• Keep track of the base; it remains the same while exponents multiply.

The power rule can dramatically reduce the number of steps needed to simplify an expression and is a fundamental principle that aids in the comprehension of more advanced algebraic operations.

Rational Exponents

Rational exponents provide an alternative to expressing roots using exponents.
A rational exponent like \( a^{m/n} \) signifies that you take the nth root of the base and then raise it to the mth power. Alternatively, you can raise it to the mth power first and then take the nth root.

• The expression \( a^{m/n} \) is equivalent to \( (\sqrt[n]{a})^m \) or \( \sqrt[n]{a^m} \).
• This notation makes it easier to write and manipulate roots in algebraic expressions.
• Rational exponents are especially useful in simplifying radical terms and carrying out operations on them.

Using rational exponents in algebra allows for smoother calculations and eases the manipulation of complex expressions involving roots.

Exponent Laws

Exponent laws are a collection of rules that simplify working with powers in algebra. They allow for predictable manipulation of terms with exponents, making it feasible to reduce expressions to simpler forms.
Some fundamental exponent laws include:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)

These laws help in systematically breaking down and simplifying expressions, ultimately making complex algebra problems more manageable. Understanding and applying these laws are crucial in algebraic manipulations and solutions.