Math Problem Statement

Solve the equation 4^(x+1) = 1/64.

Solution

The equation shown in the image is:

4x+1=1644^{x+1} = \frac{1}{64}

To solve for xx, we can rewrite both sides of the equation with the same base. Here’s the step-by-step solution:

Step 1: Rewrite 164\frac{1}{64} as a power of 4

Since 64=4364 = 4^3, we can write 164\frac{1}{64} as: 143=43\frac{1}{4^3} = 4^{-3}

So the equation becomes: 4x+1=434^{x+1} = 4^{-3}

Step 2: Set the exponents equal to each other

Since the bases are the same, we can equate the exponents: x+1=3x + 1 = -3

Step 3: Solve for xx

Subtract 1 from both sides: x=31x = -3 - 1 x=4x = -4

Final Answer

x=4x = -4

Would you like further details or have any questions about the steps?


Here are five related questions to explore this topic further:

  1. What is an exponential equation, and how do we solve equations with different bases?
  2. How can we handle exponential equations where the bases cannot be easily matched?
  3. What is the meaning of a negative exponent, and why does 43=1644^{-3} = \frac{1}{64}?
  4. How do we approach solving equations with logarithmic properties?
  5. How would the solution change if the base were different (e.g., 3x+1=1273^{x+1} = \frac{1}{27})?

Tip: When solving exponential equations, always check if you can rewrite both sides with the same base—it often simplifies the problem significantly!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Exponents
Algebra
Exponentials

Formulas

a^m = 1/a^(-m)
Exponential equation property: if a^m = a^n, then m = n

Theorems

Exponentiation rules
Properties of exponents with negative powers

Suitable Grade Level

Grades 9-11