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Chapter 4: Problem 38

Evaluate each expression if \(w=-2, x=3,\) and \(y=-1\) $$w^{-7}$$

Short Answer

Expert verified
The value of \(w^{-7}\) when \(w = -2\) is \(-\frac{1}{128}\).

Step by step solution

01

Understand the Expression

We are given the expression \( w^{-7} \) and values for \( w \), \( x \), and \( y \). Here we only need to evaluate \( w^{-7} \) where \( w = -2 \).
02

Substitute the Value of w

Replace \( w \) with \(-2\) in the expression. The expression \( w^{-7} \) becomes \((-2)^{-7}\).
03

Apply the Negative Exponent Rule

Recall that a negative exponent means taking the reciprocal of the base and changing the sign of the exponent. So, \((-2)^{-7} = \frac{1}{(-2)^7}\).
04

Calculate the Power

Now compute \((-2)^7\). Since the exponent 7 is odd, the result will be negative. Calculate \((-2)^7 = -128\).
05

Simplify the Expression

The expression \(\frac{1}{(-2)^7}\) simplifies to \(\frac{1}{-128}\).

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Key Concepts

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

Substitution
Substitution is a powerful technique used in mathematics to simplify expressions by replacing variables with their given values. Let's dive into how substitution works using the given exercise:
  • We start with the expression containing a variable: in this case, it is \( w^{-7} \).
  • The problem specifies a particular value for \( w \), which is \(-2\).
  • By substituting \( -2 \) for \( w \), the expression \( w^{-7} \) becomes \( (-2)^{-7} \).
Substitution is an essential step in solving math problems as it provides a concrete number in place of abstract variables, allowing us to proceed with further simplifications or calculations.
Negative Numbers
Understanding negative numbers is crucial when working with powers and exponents.
  • Negative numbers are numbers less than zero, often represented with a minus sign, like \(-2\).
  • When raising a negative number to a power, you must pay attention to whether the exponent is odd or even 鈥� this affects the sign of the result.
In our example, \((-2)^{-7}\), the base is a negative number, and the exponent is \(7\), which is odd. As a result, the final computation, \((-2)^7\), yields a negative value because raising a negative number to an odd power always results in a negative number.
Power of a Number
The power of a number refers to multiplying that number by itself a certain number of times, indicated by the exponent.
  • The expression \( (-2)^7 \) tells us to multiply \(-2\) by itself 7 times.
  • Calculating, we get: \((-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2) = -128\).
To interpret a negative exponent, like in \((-2)^{-7}\), you follow the rule: take the reciprocal of the base and make the exponent positive. This results in \( \frac{1}{(-2)^7} \). Applying this rule simplifies the expression to \(\frac{1}{-128}\), which is the final answer.

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