Problem 125 Simplify. Assume that all variab... [FREE SOLUTION]

91影视

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 1: Problem 125

Simplify. Assume that all variables represent nonzero integers. $$ \left\\{\left[\left(8^{-a}\right)^{-2}\right]^{b}\right\\}^{-c} \cdot\left[\left(8^{0}\right)^{a}\right]^{c} $$

Short Answer

Expert verified

\[8^{-2abc}\]

Step by step solution

Simplify the Inner Most Exponent

Simplify \[(8^{-a})^{-2}\]. According to the power of a power rule $(x^m)^n = x^{mn}$, this simplifies to \[(8^{-a})^{-2} = 8^{(-a) \times (-2)} = 8^{2a}\].

Apply the Outer Exponent

Now simplify the expression \{[8^{2a}]^b\}. Using the power of a power rule again, this becomes \[8^{2ab}.\]

Apply the Outer Most Exponent

Now we have \[\{[8^{2ab}]^{-c} = 8^{2ab \times (-c)} = 8^{-2abc}.\]

Simplify the Second Part of the Expression

Simplify the expression \[(8^{0})^{a}\]. Since any number raised to the power \{0\} is \{1\}, this becomes \[1^a = 1\]. Any number raised to the power \[c\], this stays \[1^c = 1.\]

Combine Both Parts

Now combine the two simplified parts: \[8^{-2abc} \times 1 = 8^{-2abc}.\]

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Key Concepts

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

Exponent Rules

Exponents are a way to represent repeated multiplication of a number by itself. Understanding exponent rules is essential in simplifying expressions involving powers. Here are some key rules:

Product of Powers Rule: When multiplying two powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$.
Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.
Power of a Power Rule: To raise a power to another power, multiply the exponents: $ (a^m)^n = a^{mn}$.
Zero Exponent Rule: Any non-zero number raised to the power of zero is 1: $a^0 = 1$.
Negative Exponent Rule: A negative exponent represents the reciprocal of the positive exponent: $a^{-m} = \frac{1}{a^m}$.

These rules help simplify complex expressions and solve equations involving exponents and powers.

Power of a Power Rule

The power of a power rule is one of the most used exponent rules in algebra. This rule states that when you raise a power to another power, you multiply the exponents. The general form of this rule is $ (x^m)^n = x^{mn}$. Let's apply this rule to an example:
Consider the expression: \[ (8^{-a})^{-2} \]
This can be simplified using the power of a power rule:

First, we multiply the exponents: \[8^{-a \times -2} = 8^{2a}\]

Now let's apply the rule again to the next part of the expression: \[ (8^{2a})^b \]
Using the power of a power rule again:

We multiply the exponents: \[8^{2a \times b} = 8^{2ab}\]

This rule makes it easier to simplify expressions by reducing the number of exponents you have to deal with.

Simplifying Expressions

Simplifying expressions is a crucial step in solving algebraic problems. It involves using exponent rules to reduce the expression to its simplest form. Here's a step-by-step guide to simplifying a complex expression:

Identify the innermost expression and simplify it first.
Apply the exponent rules to simplify each part.
Combine the simplified parts to get the final expression.

Let's look at our exercise again:
First, we simplify the innermost exponent:

We have \[ (8^{-a})^{-2} \]. Using the power of a power rule: \[8^{(-a) \times (-2)} = 8^{2a}\].
Next, we simplify \[ (8^{2a})^b \] to \[8^{2ab}\].
Then, we apply the outermost exponent: \[ (8^{2ab})^{-c} \] to get \[ 8^{2ab \times -c} = 8^{-2abc}\].
Now, the second part of the expression \[ (8^0)^a \] simplifies to \[1^a = 1 \], and any number raised to any power remains 1, so \[1^c = 1\].
Finally, we combine both parts: \[8^{-2abc} \times 1 = 8^{-2abc}\].

Using these steps, you can simplify any expression involving exponents.

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91影视

Simplify. Assume that all variables represent nonzero integers. $$ \left\\{\left[\left(8^{-a}\right)^{-2}\right]^{b}\right\\}^{-c} \cdot\left[\left(8^{0}\right)^{a}\right]^{c} $$

Short Answer

Step by step solution

Simplify the Inner Most Exponent

Apply the Outer Exponent

Apply the Outer Most Exponent

Simplify the Second Part of the Expression

Combine Both Parts

Key Concepts

Exponent Rules

Power of a Power Rule

Simplifying Expressions

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