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Chapter 4: Problem 15

Solve the equation. \(2^{-3 y+1}=16\)

Short Answer

Expert verified
y = -1

Step by step solution

01

- Write the given equation

The given equation is 2^{-3y + 1} = 16.
02

- Express 16 as a power of 2

Rewrite 16 as a power of 2. We know that 16 = 2^4, so the equation becomes: 2^{-3y + 1} = 2^4.
03

- Equate the exponents

Since the bases are the same, set the exponents equal to each other: -3y + 1 = 4.
04

- Solve for y

First, subtract 1 from both sides to isolate the term with y: -3y = 3. Next, divide both sides by -3: y = -1.

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Key Concepts

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

exponents and powers
Exponents and powers are essentially the shorthand notation for repeated multiplication of the same number.
For example, when we write 2^3, it means we multiply 2 by itself three times (2 * 2 * 2 = 8). The number being multiplied is called the base, and the number of times it is multiplied is called the exponent.

In the given problem, you need to express 16 as a power of 2. Knowing that 2^4 = 16 helps us in rewriting the equation, which simplifies further steps significantly.

This understanding of exponents is crucial because it allows us to solve exponential equations by equating them. It forms the basis of converting complex expressions into manageable parts.
equating exponents
When the bases in an equation are the same, you can equate the exponents to solve for the variable. In the example provided, once the bases are the same (both sides of the equation have base 2), you simply equate their exponents.

Here, the equation 2^{-3y+1} = 2^4 simplifies to solving -3y + 1 = 4 What this step essentially tells you is:
  • If 2 raised to the power a equals 2 raised to the power b, then a must equal b.

This method of solving exponential equations simplifies our work by focusing on the exponents rather than worrying about the bases.
algebraic manipulation
Algebraic manipulation involves the use of algebraic methods such as addition, subtraction, multiplication, or division to isolate the variable of interest in an equation.

In our exercise, we begin with the equation -3y + 1 = 4. To isolate y, you have to perform a series of algebraic steps:

  • First, subtract 1 from both sides to remove the constant term on the left side: -3y = 3.
  • Next, divide both sides by -3 to solve for y: y = -1.

These steps are part of basic algebraic manipulation where the goal is to isolate the variable. Being proficient in these basic techniques allows you to solve more complicated equations easily.

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