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Chapter 4: Problem 98

Solve the equation. Write the solution set with exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(8^{y^{2}-7}=64\)

Short Answer

Expert verified
y = 3 or y = -3

Step by step solution

01

Simplify the equation using exponents

Express each side of the equation as powers of the same base. Recognize that 64 is a power of 8. Since 64 = 8^2, we can rewrite the equation as 8^{y^2 - 7} = 8^2.
02

Equate the exponents

Now that both sides of the equation have the same base, set the exponents equal to each other: y^2 - 7 = 2.
03

Solve the quadratic equation

Add 7 to both sides to simplify the equation: y^2 = 9. Next, solve for y by taking the square root of both sides: y = 卤3.
04

Write the solution set

The exact solutions for y are y = 3 or y = -3. These solutions are already exact and don't need further decimal approximations.

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Key Concepts

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

quadratic equation
A quadratic equation involves expressions where the variable is squared, meaning it has the form: \[ ax^2 + bx + c = 0 \].
In our problem, after equating the exponents, we simplified to \[ y^2 - 7 = 2 \]. This can be rearranged to \[ y^2 = 9 \], a classic quadratic form \( ax^2 + bx + c = 0 \) where \( a = 1, b = 0, \text{ and } c = -9 \).
To solve this, we take the square root of both sides, resulting in \( y = \pm 3 \). This illustrates one of the simplest methods to solve a quadratic equation.
exponents and powers
Understanding exponents and powers is essential when dealing with exponential equations.
An exponent indicates how many times a number, the base, is multiplied by itself. For instance, \( 8^2 \) means \( 8 \times 8 = 64 \).
We used this concept to express 64 as \( 8^2 \), allowing us to match the exponents on both sides of the equation \( 8^{y^2 - 7} = 8^2 \).
solution sets
A solution set is a collection of all possible solutions that satisfy the given equation.
For the quadratic equation \( y^2 = 9 \), the solutions are \( y = 3 \) and \( y = -3 \), forming a solution set \( \{ 3, -3 \} \).
Always double-check your solution set to ensure that all values satisfy the original equation.

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Most popular questions from this chapter

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