Problem 89 - Algebra and Trigonometry

Question

Evaluate each expression without using a calculator. $$ 32^{-\frac{4}{5}} $$

Step-by-Step Solution

Answer

So, \(32^{-\frac{4}{5}}\) equals \(\frac{1}{16}\).

Step 1: Convert the Negative Exponent to a Reciprocal

When an expression is raised to a negative exponent, it implies the reciprocal of the base. So, the first step is to convert \(32^{-\frac{4}{5}}\) to \(\frac{1}{32^{\frac{4}{5}}}\).

Step 2: Break down the Fractional Exponent

Recall that a fractional exponent is seen as the root of a number. In this case, when \(\frac{4}{5}\) is the exponent, it implies the 5th root of the base to the power of 4. Transforming \(\frac{1}{32^{\frac{4}{5}}}\) gives \(\frac{1}{{(5\sqrt{32})}^4}\).

Step 3: Simplify the Base

The base 32 can be simplified as 2 raised to the power of 5: \(2^5 = 32\). Utilizing this, the expression \(\frac{1}{{(5\sqrt{32})}^4}\) now becomes \(\frac{1}{{(5\sqrt{2^5})}^4}\).

Step 4: Simplify the Expression

Simplify the base further: \(\frac{1}{{(5\sqrt{2^5})}^4} = \frac{1}{{(2^{5/5})}^4} = \frac{1}{{2^4}}\). The term \(2^4\) means \(2*2*2*2\), which equals 16.

Key Concepts

Negative Exponents

Negative exponents can be a puzzling concept at first, but they're quite simple once you understand the basic idea. A negative exponent indicates that the base should be taken as its reciprocal.
For example, if you have a number raised to a negative exponent such as \( a^{-n} \), it is equivalent to \( \frac{1}{a^n} \).
This means you flip the base to turn the exponent positive. - Understanding negative exponents is crucial because they show up frequently in algebra and calculus.
They simplify complex-looking expressions and help solve equations more efficiently.
Whenever you see a negative exponent, remember: "flip the base to make it positive!"

Fractional Exponents

Fractional exponents provide a powerful way to express roots and powers in a single term. They are seen as an extension of both concept of roots and powers.
If you have a number with a fractional exponent like \( a^{m/n} \), it can be interpreted as \( \sqrt[n]{a^m} \), which is the \( n^{th} \) root of \( a \) raised to the power of \( m \).
- The denominator of the fraction, \( n \), indicates the type of root, while the numerator, \( m \), represents the power.
This dual nature allows you to simplify roots and powers in one elegant step.
By combining roots and powers, fractional exponents help you perform calculations easily and neatly without separate root and power steps.

Exponent Laws

Exponent laws are vital for simplifying expressions involving powers. These laws enable you to manipulate expressions easily by adhering to a few basic rules.
Here are some handy rules to remember:

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

These rules are the building blocks of algebraic manipulation.
They make complex expressions simpler, allowing equations to be solved with ease.

Evaluating Expressions

Evaluating expressions involves calculating or simplifying the given expression to its simplest form. This often requires applying a mix of arithmetic operations and algebraic rules, including exponent laws.
When evaluating an expression like \( 32^{-\frac{4}{5}} \), you need to:

• Apply the negative exponent rule to convert it to a reciprocal.
• Break down the fractional exponent to interpret roots and powers.
• Simplify further using exponent laws and known values (e.g., \( 32 = 2^5 \)).
• Perform arithmetic operations to arrive at the final result.

Knowing how to evaluate expressions correctly strengthens your problem-solving skills.
It also allows you to understand the relationships between numbers and operations more deeply.
Always take it step by step, and make use of exponent rules to guide the process.