Problem 68 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ \frac{(9 s t)^{3 / 2}}{\left(27 s^{3} t^{-4}\right)^{2 / 3}}\left(\frac{3 s^{-2}}{4 t^{1 / 3}}\right)^{-1} $$

Step-by-Step Solution

Answer

The simplified expression is \(12s^{3/2}t^{29/6}\). There are no negative exponents.

Step 1: Simplify the Exponent of the Numerator

The expression given is \( \frac{(9st)^{3/2}}{(27s^3t^{-4})^{2/3}} \left(\frac{3s^{-2}}{4t^{1/3}}\right)^{-1} \). First, expand \((9st)^{3/2}\). This equals \(9^{3/2} \cdot s^{3/2} \cdot t^{3/2}\). Note that \(9^{3/2} = \sqrt{9^3} = 27\). Thus, the numerator is \(27s^{3/2}t^{3/2}\).

Step 2: Simplify the Denominator

Now, simplify \((27s^3t^{-4})^{2/3}\). Start by taking each component:- \(27^{2/3} = \sqrt[3]{27^2} = 9\) (since \(27^{2/3} = (3^3)^{2/3} = 3^2 = 9\)).- \((s^3)^{2/3} = s^2\).- \((t^{-4})^{2/3} = t^{-8/3}\). Thus, the denominator simplifies to \(9s^2t^{-8/3}\).

Step 3: Simplify the Last Term with Reciprocal

Next, address \(\left(\frac{3s^{-2}}{4t^{1/3}}\right)^{-1}\). Inverting this fraction, we get \(\frac{4t^{1/3}}{3s^{-2}}\). This simplifies to \(\frac{4t^{1/3}s^2}{3}\) as \(s^{-2}\) moves to the numerator as \(s^2\).

Step 4: Combine and Simplify the Entire Expression

Now combine all simplified expressions:- Numerator becomes \(27s^{3/2}t^{3/2} \cdot 4t^{1/3}s^2 = 108s^{7/2}t^{11/6}\).- Denominator remains \(9s^2t^{-8/3}\).Divide the expressions: - Simplify \(\frac{108}{9} = 12\).- Simplify the exponents: - \(s^{7/2} \div s^2 = s^{3/2}\). - \(t^{11/6} \div t^{-8/3} = t^{11/6 + 8/3} = t^{29/6}\).

Step 5: Final Expression

After simplification, the final expression is \(12s^{3/2}t^{29/6}\). There are no negative exponents.

Key Concepts

Exponent Rules

Understanding exponent rules is fundamental in simplifying algebraic expressions. Exponents indicate how many times a number, known as the base, is multiplied by itself. Here are some basic rules:

• Product of powers: When multiplying expressions with the same base, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a power: When raising an expression with an exponent to another power, multiply the exponents. For instance, \( (a^m)^n = a^{m\cdot n}\).
• Power of a product: Distribute the exponent to each factor in the product. Thus, \( (ab)^n = a^n \cdot b^n\).
• Quotient of powers: When dividing expressions with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n}\).

Understanding these rules helps simplify expressions more efficiently and eliminate negative exponents, as seen in the original exercise.

Fraction Simplification

Fraction simplification can make complex expressions more manageable. The core idea is to reduce the fraction by eliminating common factors in the numerator and denominator.
For example:

• Divide both the numerator and the denominator by their greatest common factor.
• In algebra, also look for and simplify variables. For instance, if you have \(\frac{x^m}{x^n}\), simplify this to \(x^{m-n}\), assuming \(m > n\).

In the exercise, simplifying \(\frac{108s^{7/2}t^{11/6}}{9s^2t^{-8/3}}\) involved dividing each by the highest factors and subtracting the exponents to simplify further. The fraction \(\frac{108}{9}\) simplifies to 12, streamlining the entire expression.

Negative Exponents

Negative exponents can be confusing but are easy to understand with the right approach. A negative exponent means to take the reciprocal of the base and make the exponent positive.
Here's how to deal with them:

• For \(a^{-n}\), this equals \(\frac{1}{a^n}\).
• Conversely, \(\frac{1}{a^{-n}} = a^n\).

In the exercise, \(t^{-8/3}\) appeared in the denominator. Applying the negative exponent rule, we handle it as \(\frac{1}{t^{8/3}}\), simplifying to multiply into \(t^b\). This approach gets rid of the negative exponent.

Rational Exponents

Rational exponents represent roots and powers in a single expression, connecting exponents to roots. It has the form of a fraction, with the numerator as the power and the denominator as the root.
For instance:

• \((a^{m/n})\) means you raise \(a\) to the power of the numerator \(m\) and take the nth root: \(\sqrt[n]{a^m}\).
• This rule converts traditional roots to exponent form, blending multiplication and root extraction.

In the exercise, \((9st)^{3/2}\) was simplified using these rules: \(9^{3/2} = \sqrt{9^3}\), and results in more simplified integers and variables, eliminating complex roots in the process. Recognizing this combination helps tidy up seemingly tangled algebraic expressions quickly.