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Chapter 7: Problem 84

In Exercises \(83-88\), evaluate the given expression for \(x=2\) and \(y=-3\) $$y^{-2}$$

Short Answer

Expert verified
The value of \(y^{-2}\) for \(y=-3\) is \(\frac{1}{9}\).

Step by step solution

01

Identify the given values

The given values are \(x=2\) and \(y=-3\). The expression to evaluate is \(y^{-2}\).
02

Substitute the value of \(y\)

Substitute \(y=-3\) into the expression \(y^{-2}\), giving \((-3)^{-2}\).
03

Apply the negative exponent rule

Recall that \(a^{-b} = \frac{1}{a^b}\). Hence, \((-3)^{-2} = \frac{1}{(-3)^2}\).
04

Evaluate the exponent

Calculate \((-3)^2\), which is 9. Therefore, \((-3)^{-2} = \frac{1}{9}\).
05

Simplify the expression

The simplified result of the expression is \(\frac{1}{9}\).

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Key Concepts

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

negative exponents
Negative exponents may seem tricky, but understanding them can make algebra much simpler. A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For instance, if you have an expression like \(a^{-b}\), this is equivalent to \(\frac{1}{a^b}\). This algebraic rule flips the base and turns the exponent positive. In our exercise, \(y^{-2}\) means finding the reciprocal of \(y\) squared. First, we substitute \(-3\) for \(y\). So, \((-3)^{-2}\) becomes \(\frac{1}{(-3)^2}\). This reciprocal rule helps us solve many similar algebraic problems efficiently.
substitution
Substitution is a fundamental technique in algebra where we replace a variable with a given number. In our exercise, we are provided with specific values \(x=2\) and \(y=-3\). The task is to evaluate \(-3\) to the power of \(-2\). When substituting, ensure accuracy: one wrong substitution can change the entire result. So, substituting \(y = -3\) into \(y^{-2}\), we get \(-3^{-2}\). Now, the problem turns from an abstract expression into something numerical we can work with.
simplification
Simplification is about breaking down complex expressions into simpler components that are easier to work with and understand. By converting negative exponents to their reciprocal forms, we simplify our calculations. From the exercise, \((-3)^{-2}\) uses the rule \(a^{-b} = \frac{1}{a^b}\). Applying this, it becomes \(-3^{-2} = \frac{1}{(-3)^2}\). Next, we simplify further by calculating the square of \(-3\). Since \(-3) (-3) = 9\), our expression becomes \(\frac{1}{9}\). Now, we have a simple fraction that represents our final answer.
algebraic rules
Understanding and applying algebraic rules is crucial for solving expressions correctly. In this exercise, we've used several key rules. First, the rule for negative exponents \(a^{-b} = \frac{1}{a^b}\). This transformation made it possible to convert a tricky-looking negative exponent into a straightforward fraction. Second, we applied the exponentiation rule for squaring a number, which means multiplying the base by itself. Lastly, throughout the solution, we followed consistent substitution and simplification steps. Mastering these basic rules provides a strong foundation for tackling more complex algebraic problems in the future.

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Most popular questions from this chapter

What can you say about the sign of the exponent of 10 of a number written in scientific notation if the number is bigger than \(1 ?\) If the number is smaller than \(1 ?\) Why?

Convert each number into standard notation. $$0.026 \times 10^{-3}$$

Do your computations using scientific notation. If one atom of iron has a mass of \(9.3 \times 10^{-23}\) gram, what is the mass of \(80,000\) atoms?

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 a-5)^{2}$$

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