Problem 34 - Algebra 2

Question

Simplify each number. $$(32)^{-\frac{4}{5}}$$

Step-by-Step Solution

Answer

The simplification of \( (32)^{-\frac{4}{5}} \) is \( \frac{1}{16} \).

Step 1: Apply the exponential rule

Using the rule of exponentials, change \( (32)^{-\frac{4}{5}} \) to \( \frac{1}{32^\frac{4}{5}} \)

Step 2: Take the fifth root and raise it to power 4

The expression reads as 'the fifth root of 32, raised to the power 4'. The fifth root of 32 is 2, thus the expression simplifies to \( \frac{1}{2^4} \).

Step 3: Calculate the power

Calculate 2 to the power of 4, which results in 16. So the simplification is \( \frac{1}{16} \).

Key Concepts

Understanding Exponential Rules

Exponential rules are mathematical principles that help us manipulate expressions involving exponents, making them easier to work with. One key exponential rule is that a negative exponent indicates a reciprocal. For example, \( a^{-n} = \frac{1}{a^n} \). This means if you have a negative exponent, you can convert it to a positive one by taking the reciprocal of the base raised to the positive exponent. This is crucial in simplifying expressions that might initially look complex. Another rule is the power of a power rule, which states \( (a^m)^n = a^{m \cdot n} \). Knowing these rules not only simplifies calculations but also enhances your math problem-solving skills.

These rules are like tools in a toolkit. You can apply them to break down and solve various expressions consistently. Start by examining the exponents and see which rule fits your expression best.

Simplifying Expressions with Exponents

Simplifying expressions involving exponents can transform a complicated-looking problem into something much simpler. It involves reducing the expression to its most concise form while maintaining its value. Let's consider \( (32)^{-\frac{4}{5}} \) as an example. By applying exponential rules, we simplify by converting it first into its reciprocal, which is \( \frac{1}{32^{\frac{4}{5}}} \). This step makes further simplifications easier to manage.

To continue simplifying, read the expression as operations on numbers rather than just symbols. Think of each part as a distinct action you need to take, and follow the rules accordingly. Using the properties of exponents, especially when dealing with roots and powers, leads to simplicity from apparent complex expressions. By doing this, not only do you reduce the risk of errors, but you also understand the numbers more profoundly.

The Concept of Fifth Root

The fifth root of a number refers to a value that, when raised to the power of five, equals the original number. It's the opposite of raising a number to the fifth power. Calculating roots can be tricky, but recognizing patterns and knowing certain common roots helps.

In \( (32)^{-\frac{4}{5}} \), after shifting from a negative exponent to a reciprocal, the focus moves to \( 32^{\frac{4}{5}} \). This means we first find the fifth root of 32, which is 2, since \( 2^5 = 32 \). Understanding roots like this is helpful, especially in simplifying expressions where both roots and powers are involved.

• Recognizing common roots speeds up calculations.
• Being comfortable with roots assists in various algebraic manipulations.

By navigating through roots comfortably, you can focus more on the arithmetic and less on figuring out what the terms mean.

Dealing with Negative Exponents

Negative exponents often seem daunting, but they are simple once understood. A negative exponent suggests taking the reciprocal of the base raised to the corresponding positive power, as seen in \( (32)^{-\frac{4}{5}} = \frac{1}{32^{\frac{4}{5}}} \). This rule is crucial because it offers a straightforward way to rewrite and simplify expressions involving negative exponents.

The logic behind this rule is based on the continuity of multiplication and division in mathematics where multiplying by a negative power means dividing repeatedly. This reciprocal approach simplifies expressions and calculations, making it easier to solve complex problems.

• Negative exponents don't mean the result is negative.
• They indicate inversion or reciprocal operations.

This understanding demystifies negative exponents, turning a potential stumbling block into an opportunity for effective problem solving.