Problem 329 In the following exercises, simp... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 329

In the following exercises, simplify. (a) $(-8)^{-2} \quad$ (b)$-8^{-2}$

Short Answer

Expert verified

Simplified, $ (-8)^{-2} = \frac{1}{64} $ and $ -8^{-2} = -\frac{1}{64} $.

Step by step solution

Title - Understanding negative exponents

Negative exponents indicate that the base should be taken as the reciprocal. For any base 'a' and exponent '-n', the expression $ a^{-n} $ is equal to $ \frac{1}{a^n} $.

Title - Simplify $ (-8)^{-2} $

Given $ (-8)^{-2} $, apply the property of negative exponents to simplify: $ (-8)^{-2} = \frac{1}{(-8)^2} $. Next, calculate $ (-8)^2 $, which is $ 64 $. Therefore, $ (-8)^{-2} = \frac{1}{64} $.

Title - Simplify $ -8^{-2} $

Given $ -8^{-2} $, the negative sign before the base remains, and now consider the exponent: $ 8^{-2} = \frac{1}{8^2} $. Calculate $ 8^2 $, which is $ 64 $. So, $ 8^{-2} = \frac{1}{64} $. Therefore, $ -8^{-2} = -\frac{1}{64} $.

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Key Concepts

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

Reciprocal of Base

When dealing with negative exponents, one crucial concept to understand is the reciprocal of the base. A negative exponent means you need to flip the base into its reciprocal form.
When you see something like $a^{-n}$, it signifies that you need to take 1 divided by $a$ raised to the positive exponent $n$.
For instance, $(-8)^{-2}$ can be rewritten as $\frac{1}{(-8)^2}$.
Here, $a = -8$ and $n = 2$.
1. Reciprocate the base: Instead of just $-8$, we construct $ \frac{1}{(-8)^2} $.
2. Solve the reciprocal: Then calculate $(-8)^2 = 64$ to get $\frac{1}{64}$.
This is the key step to transforming and simplifying expressions with negative exponents.

Exponentiation

Exponentiation means raising a base number to a power. The power tells us how many times the base number is multiplied by itself.
In the expression $(-8)^2$, the base is $-8$ and the exponent is $2$.
Here鈥檚 how to handle exponentiation:

Identify the base and exponent: Base $-8$, Exponent $2$.
Multiply the base by itself $n$ times: Calculate what $(-8) \times (-8)$ equals.

When calculated, $(-8)^2 = 64$.
Exponentiation is particularly important in simplifying expressions with negative exponents since it initially provides us with a value in its positive exponent form before simplifying further.

Simplify Expressions

Simplification makes expressions easier to understand and work with. Here鈥檚 a step-by-step approach:

Convert negative exponents by taking reciprocals: $(-8)^{-2}$ becomes $\frac{1}{(-8)^2}$.
Calculate exponentiation first: Compute $(-8)^2 = 64$.
Divide 1 by the result of exponentiation: $\frac{1}{64}$

In context, here's how we go from $8^{-2}$ to simplified form:
First, understand that $8^{-2}$ is actually $\frac{1}{8^2}$.
Next, calculate $8^2 = 64$. The simplified form is then $\frac{1}{64}$.
For $-8^{-2}$, keep the negative sign intact, compute $8^{-2} = \frac{1}{64}$, and finally, denote it as $-\frac{1}{64}$.
Simplification helps break down and resolve complex exponents step by step, leading to clearer and more manageable expressions.

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91影视

In the following exercises, simplify. (a) \((-8)^{-2} \quad\) (b)\(-8^{-2}\)

Short Answer

Step by step solution

Title - Understanding negative exponents

Title - Simplify \( (-8)^{-2} \)

Title - Simplify \( -8^{-2} \)

Key Concepts

Reciprocal of Base

Exponentiation

Simplify Expressions

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