Problem 78 Simplify each expression. Assume... [FREE SOLUTION]

Chapter 10: Problem 78

Simplify each expression. Assume that all variables represent positive real numbers. $$ \frac{a^{-1 / 2} b^{-5 / 4}}{\left(a^{-3} b^{2}\right)^{1 / 6}} $$

Short Answer

Expert verified

$ \frac{1}{b^{19/12}} $

Step by step solution

- Simplify the Denominator

First, simplify $(a^{-3} b^{2})^{1/6}$. Use the exponentiation rule ($ (x^m)^n = x^{mn} $). So, $ (a^{-3} b^{2})^{1/6} = a^{-3 \times 1/6} b^{2 \times 1/6} = a^{-1/2} b^{1/3} $.

- Rewrite the Expression

Rewrite the original expression using the simplified denominator from Step 1: $ \frac{a^{-1/2} b^{-5/4}}{a^{-1/2} b^{1/3}} $.

- Apply the Quotient Rule

Apply the quotient rule for exponents ($\frac{x^m}{x^n} = x^{m-n}$). Combine the exponents for $a$ and $b$: $ a^{-1/2 - (-1/2)} b^{-5/4 - 1/3} = a^{0} b^{-5/4 - 1/3} $.

- Simplify the Exponents

Since $a^{0}$ equals 1, the expression simplifies to $b^{-5/4 - 1/3}$. To add $-5/4$ and $-1/3$, find a common denominator and simplify: $ -\frac{15}{12} - \frac{4}{12} = -\frac{19}{12} $. So, the expression becomes $ b^{-19/12} $.

- Simplify the Final Expression

Rewrite the expression with a positive exponent: $ b^{-19/12} = \frac{1}{b^{19/12}} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponentiation Rules

Exponentiation is a fundamental math concept that involves raising numbers or variables to a power. Let's break it down simply. If you have an expression like $a^m$, it means multiply $a$ by itself $m$ times. Some key exponentiation rules include:

Product Rule: $ x^m \times x^n = x^{m+n} $. This allows you to add the exponents if the bases are the same.
Power of a Power Rule: $(x^m)^n = x^{mn} $. This means you multiply the exponents when taking a power of a power.
Power of a Product Rule: $(xy)^m = x^m y^m $. Apply the exponent to both bases inside the parenthesis.
Zero Exponent Rule: $x^0 = 1 $. Any base raised to the zero power is one.
Negative Exponent Rule: $x^{-m} = \frac{1}{x^m} $. A negative exponent suggests the reciprocal.

Understanding these rules makes it easier to simplify complex expressions, as seen in the solution steps where these rules simplify the denominator and numerator of our original fraction.

Quotient Rule for Exponents

The Quotient Rule for Exponents is essential when dividing expressions with the same base. It states: $\frac{x^m}{x^n} = x^{m-n}$. Essentially, you subtract the exponent in the denominator from the exponent in the numerator.
This rule was applied in Step 3 of the solution to the given problem when dealing with the expression $\frac{a^{-1/2} b^{-5/4}}{a^{-1/2} b^{1/3}}$. Here's a more detailed look:

Same Base Simplification: For $a$, you have $a^{-1/2}$ and $a^{-1/2}$. Subtracting the exponents ($-1/2 - (-1/2)$) gives us $a^0 = 1$. This shows how it simplifies.
Subtract Exponents: For $b$, you have $b^{-5/4}$ in the numerator and $b^{1/3}$ in the denominator. This means you subtract $1/3$ from $ -5/4$, resulting in $b^{-5/4 - 1/3}$. As a step-by-step, find the common denominator (which is 12 here), and subtracting gives $-\frac{19}{12}$.

This rule simplifies complex rational expressions, making them clear and elegant.

Simplifying Exponents

To simplify exponents means to make expressions with powers easier to understand and work with. Breaking down and combining exponents step-by-step is crucial.
Here's how you can do it:

Combining Like Terms: Look for terms with the same base. Use the Product Rule or Quotient Rule to combine them into a single term.
Negative Exponents: To handle negative exponents, apply the Negative Exponent Rule. Reverse the term and make the exponent positive.
Fractional Exponents: When exponents are fractions, remember they represent roots. For example, $a^{1/2} = \sqrt{a} $.

In our specific exercise, after combining the exponents and simplifying the terms for both the numerator and denominator, we ended up with a negative fractional exponent, $b^{-19/12}$. To simplify, rewrite it using the Negative Exponent Rule, resulting in $ \frac{1}{b^{19/12}} $. Simplifying exponents transforms complex algebraic expressions into manageable pieces, providing clarity and an efficient solution path.

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91影视

Simplify each expression. Assume that all variables represent positive real numbers. $$ \frac{a^{-1 / 2} b^{-5 / 4}}{\left(a^{-3} b^{2}\right)^{1 / 6}} $$

Short Answer

Step by step solution

- Simplify the Denominator

- Rewrite the Expression

- Apply the Quotient Rule

- Simplify the Exponents

- Simplify the Final Expression

Key Concepts

Exponentiation Rules

Quotient Rule for Exponents

Simplifying Exponents

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